Why would a grad school not want me to defer admission?
The MIDS program at Berkeley is an unusual special case among graduate programs. One clue is the use of admissions counselors, which are almost unheard of in typical graduate programs but are widely used in for-profit education. (This doesn't mean there's anything intrinsically problematic about relying on admissions counselors. They are used because they are an effective recruiting technique and can genuinely help everything work more smoothly for applicants. However, it's a cultural difference in how recruiting is done.)
In fact, the MIDS program is run by the I School at Berkeley in partnership with a for-profit company called 2U. See here for a description of how 2U and Berkeley are collaborating on this. In particular, the key sentence is "2U's marketing and recruiting team supports the I School's goal of finding students who are most likely to succeed in the rigorous Master of Information and Data Science program." The admissions counselor you've worked with is presumably a 2U employee from the marketing and recruitment team, whose job performance is based on getting people to enroll. Deferring at least means you won't be enrolling now, and you might change your mind about enrolling at all in the meantime, so it would not be considered a good outcome by 2U.
So why is Berkeley working with 2U if it could lead to awkward situations like this? One reason is 2U's educational technology platform, but frankly I think it's primarily the prospect of making money. The I School is descended from Berkeley's library school, and this has never been a lucrative field. I'm sure the idea of running a professional master's program (in which people's employers pay a lot of money) is very attractive to the I School, especially if 2U takes care of recruiting students and managing the online program itself. This seems to be a common pattern in 2U's partnerships with universities.
This situation actually gives you a little more leverage than you might normally have:
Feel free to be firm (but polite, of course) with the admissions counselor. This is someone who is trying to sell you something, and you don't need to worry that they have some wise but inscrutable motivation for trying to get you to enroll now.
The I School really doesn't want 2U to embarrass them. If you have a bad experience with 2U, I'm confident that the I School would like to know and will try to keep 2U in line.
In the worst case scenario, you could try complaining to the Berkeley administration (outside of the I School), or even doing something like bringing problems to the attention of someone at the Daily Californian. I'm sure many people at Berkeley dislike the idea of partnering with a for-profit company, and this gives the I School and 2U a strong incentive to address complaints effectively.
It sounds to me like you have a legitimate complaint. Either the admissions counselor is trying to keep you from exercising a legitimate option, or they misled you about what your options would be. It doesn't sound like a simple miscommunication, since the counselor could have easily addressed that by providing the correct information or context. It's not clear how things will play out, but you shouldn't give up just because the admissions counselor is uncooperative.
I think there is no point in our speculating about the motivation behind the "aggressive pushback". In that we don't even know which program you're talking about, we have less information than you. Moreover, speaking as an academic at an "analogue" university, I have no experience whatsoever with online programs. Maybe someone here will have expertise in this area, but I can't think of any frequent contributor with that background.
The counselor did mention at one point that I could defer admission by one trimester, should I be accepted.
Did you get that in writing (presumably email)? If so, I would send a copy of this offer back to your "admissions counselor" and then have one more telephone conversation with him in which you do not ask for advice at all but simply say "I would like to exercise the deferral option that you previously offered." If they try to argue, don't. Just say, "I'm sorry if I wasn't making myself clear: I am not interested in further discussion on this point. I would like to take you up on your previous offer. Who should I talk to about doing that?"
If you didn't get the offer in writing: well, the bottom line is that I don't see how you can hold them to it. You should decide whether you are interested in enrolling in the program without the deferral. It's not my decision, for what it's worth: this kind of behavior on the part of their organization makes a poor impression on me. It is not a good foundation for future work. Unless you feel like this is a unique opportunity (and, well it sounds snooty but nevertheless: I don't know of any online graduate programs that are unique opportunities), I would not take them up on their modified offer. If you decide that a deferral is not okay, you have a version of the previous conversation where instead of sending the documentation you indicate that you were extended the offer before. The option is now theirs: they can enroll you after one trimester or not at all.
I am very conscious that the above is more like how I would deal with my credit card company than with a university. Again, I would think twice about enrolling in a graduate program that reminds anyone of a credit card company: yikes.
Good luck.
Added: The OP has since provided two further significant pieces of information. First, the institution is one of the leading ones in the US; I was worried that it was an online only institution. Second, there are three application cycles per year, so really it is hard for me (a tenured faculty member at a nationally ranked state research university) to understand what the big deal is to start in the following cycle.
Here is my updated advice: as above, I still recommend that you have one more telephone conversation in which you make a good faith effort to get what you want out of the person who's been assigned to deal with you. From your description of the previous conversations, it doesn't sound like you've been as direct as you possibly could be. Counseling is over. Either this person will arrange your deferral or he won't: find out which it is. If the conversation with him does not result in your successful deferral, you should contact a faculty member in the program and tell them that you want to start in the next trimester, and that [insert Mr. Counselor's name here] orally gave you the option of doing that. If there is a good reason not to do this, you'll hear it much more directly from the people who are actually running the program than an "admissions counselor" (again, I don't really know what that is...). I looked up the program you linked to, and three minutes of internet searching did not fully explicitly tell me which faculty member to contact about questions like this, but there is one faculty member who appears most prominently on the page, namely the dean of the school you've applied to. I recommend that you write to her.
Like Pete L. Clark, I don't have a lot of experience with online programs, but I can extrapolate some guesses from what I know about traditional "face-to-face" graduate programs.
In many traditional programs, it would be quite unusual to offer a student the option to defer for less than a full year. The courses required for the degree often are not offered every term, and may have to be taken in a particular sequence, so the department will schedule courses so that students who start at the beginning of an academic year can take everything they need in sequence and be done as quickly as possible. For a student who defers and starts mid-year, there is a problem. Either the student has to wait to take some of their courses (complicating their schedule or even delaying their graduation), or they have to take courses out of sequence (problematic if one course develops material that is prerequisite for the next), or the department has to offer a special extra course (expensive) or arrange for something like independent study (awkward and time-consuming).
An online program might be able to mitigate this issue, if some of the courses can be taken "asynchronously" (so that you don't have to be "taking" the course at the same time as a professor is "teaching" it). But this is not always the case; sometimes online course are quite synchronous and resemble traditional courses in structure, except that lectures and assignments are delivered electronically.
So that's at least one reason why they might not want you to defer, and why they might not be able to "just let you do what is best for you".
So why were you offered this option in the first place? Another note: traditional graduate programs are usually run "in-house" by a department and its professors, with administrative assistance from department staff. Admissions is usually handled that way as well (in contrast to undergraduate programs, for which there is a dedicated university-wide admissions office). It would be unusual for there to be dedicated admissions counselors for a graduate program; usually the point of contact for prospective students would be a professor from the department, or a staff member with a direct line to a professor. The fact that all your dealings have been with a person whose sole job appears to be admissions suggests that this person is somewhat removed from the day-to-day running of the graduate program. So they may have mistakenly offered you this option without checking with the department, whose responsibility it would be to actually make it work. It's conceivable that the department actually doesn't want to have this option available (perhaps for the scheduling reasons described above), and told the admissions counselor so, and now he/she is trying to backtrack.
As such, here are some things I might suggest:
Try to get in touch with someone who is actually involved in running the program. The academic department which offers the program (say, computer science) should have a professor listed on their web site as "graduate chair" or some such title. That would be the ideal person. See if you can find out from them what would actually be involved (for you and for them) if you were to defer. This may at least give you a better understanding of what's going on, and you can have a sensible discussion with someone, instead of a credit card company-style ultimatum.
Based on your work situation, would it be convenient for you to defer for an entire year? My guess is that the university would be much better able to accommodate that. It does cause some issues with "balancing cohort size" and revenue. (More specifically, they probably have a fixed maximum number of students that can enroll in each year. If you defer, they will be (further) below that number for this year, which represents lost tuition revenue that can't necessarily be made up by increasing enrollment next year.) But that is likely a much smaller logistical problem than a student starting mid-year.