To break mental health condition to potential new advisor?

First, congratulations on taking that first step towards recognizing yourself for who you are and seeking assistance to change yourself and get closer to becoming the person you would prefer to be. That is a huge step that many people are not willing to take, and I must applaud you for your courage.

Now, the issue at hand is whether you should reveal your current state of mind to your advisor. That's a delicate and deeply personal issue which can really only be left to your choice. That being said, if I were in your shoes, I would do so. In fact, I did exactly just that, both to my advisor during school and all my supervisors at work, and I benefited greatly from it. (Note that this is personal experience. If your advisor's personality is… less than accommodating, you may have a different experience.)

One of the most important roles of an advisor is to make sure that their students stay on track, both in terms of the scope of their research and their timelines. The last thing a professor wants is to be an inadequate advisor and as a result have a student take longer to graduate or turn out mediocre papers. Of course, this does depend on how busy the professor is, and they may ask other faculty for help or ask you to be mostly responsible for your own work (the last of which is relatively rare but not unheard of).

Psychological issues can cause many problems, including motivation. You can only benefit from having external motivation to keep you from deviating from your intended timeline, especially from someone you respect. At the same time, having that someone understand that you have difficulties will release you from the pressure of wanting to not disappoint that person, which ironically would have possibly caused you to stress out and as a result end up disappointing that person, then get stressed out from that outcome in a vicious cycle. Inform your advisor of your issues, and then together you can work out how to best manage your workflow. Perhaps you might need more frequent reminders, or require a little more detail with advice/instructions. Most advisors that I know love being helpful to their students, and hopefully your advisor will be the same.

Please note that you wouldn't be using your current state of mind as an excuse. You are already self-aware of your issues, which means that you will recognize when you attempt to avoid doing something and are capable of changing the course of that behavior when it occurs. Having that external motivator could be the extra push you need to help you turn away from avoidance and back to getting things done.

As for the issue of disclosing your psychological condition, it is not required unless it is directly relevant to the matter at hand. In fact, as mentioned in the answer to the question you linked, it can be considered unethical to unnecessarily disclose such information. If your adviser does find it necessary, let them know about your concern and work together to word it so that it minimizes any undue negative connotations.

I don't have first-hand experience with mental/psychological difficulties, but I did have a dear friend with such (but more severe) issues while we were both pursuing our doctorates. I'll tell you what I told him when he asked me the same question: no, don't tell your supervisor that. At least not at the very beginning.

First of all, there's no reason to do this in one sitting, if you really want to do it. You will have lots of meetings with your supervisor, and many chances to divulge little pieces of information about your state here and there. I can't really tell you how to do that gradually though. That's something that you'll need to figure out for yourself (assuming you want to).

Next, think about pros and cons. What is the best case scenario? How might telling your supervisor ultimately help you? I think you'll find that even if your supervisor is a very accommodating and understanding person, there isn't much to gain by telling him/her. What is likely to happen is that you'll get a free pass when things get tough. Do you want that? If, at some point, things do get extremely tough and you need a little space, talk to your supervisor then.

On the other hand, and without more information to go on, there's a bunch of things that might go awry. Things might get a little awkward between you, perhaps because the supervisor is a bit inept and without any experience of her/his own; (s)he might think that perhaps you are asking for a free pass, when in reality you are perfectly fine, etc.

I'm probably going to be one of the very few who advocate not telling your supervisor; if you want to take something from my answer, let it be this: don't do anything abrupt. Test the waters first. And weigh pros and cons. After all, you seem to be fully aware of your issues. Plus, you are seeking outside help.

No hurry in sharing everything in the first meeting. Most advisors meet with their students weekly anyway -- certainly mine did -- and you can ask, "Shall I come see you once a week to discuss my progress?" with an eager expression. (If he says no, don't give up -- you may be able to steer things that way later, and you may form a study group that meets at least once a week.)

That said, if some or all of your health vicissitudes happen to come up in conversation tomorrow, that's okay too.

It's exciting to hear about the two types of support you have. In addition, you may want to ask your treating therapist, or a special evaluator, or your primary care provider, for the needed documentation to establish yourself formally with your university's office for students with disabilities. This is a great way of carrying an umbrella in case of rain, and of doing self-care. You never know when having this recognition may come in handy.