PhD thesis defense - is there a separate defense session with one's graduate committee?

That varies from place to place, even in the US. Your advisor will give you the best advice. But yes, a private session with your committee is pretty common. They want to assure themselves that you are "seasoned" and have both the required depth and breadth to be awarded the degree.

Again, your advisor will tell you what to expect, but it is possible to get questions that are unrelated to your thesis, or even to your specific subfield. In mathematics, members of a committee often have different specialties and may want to know what you can say about their own field as they may have only a sketchy knowledge of yours and of your thesis.

Don't, in general, expect it to be just a presentation. The public part needs to be a bit less technical in tone, since many of those present are looking for an overview. That isn't enough, of course. You need to say what you did and the key ideas that got you there.

Each experience is different. In my experience, the entire defense was public. The committee asked questions throughout the presentation, then more at the end if they had any others. There was also a fair bit of discussion and asking my thoughts on potential research directions. After this, they offered anyone in the audience the opportunity to ask questions. After everyone was satisfied, then they asked people to step out (including me) to briefly step out. When they opened the doors they announced results.

Each committee chair may do this differently but my experience was as follows:

1. Ask everyone (including candidate) not on committee to step out briefly. Doors close.
2. Candidate and audience brought back in.
3. Defense presentation (questions during).
4. Questions after by committee.
5. Questions from audience (if time permits).
6. Everyone not on committee (including candidate) step out. Doors close.
7. Doors open. Result announced.

I will answer in the context of my PhD defense and the defenses of my students, who are in various versions of public health.

There is a separate session with the graduate committee. This can last a long time, or it can be fairly brief, and in my experience, there's not a strong correlation with length of time and success.

In that session, somewhat harder questions, ambiguous questions, etc. can be asked. What would you do in X circumstance. Push the conclusions of your study a little farther. On occasion, and I dislike it when this happens, there's objections from one or more faculty members about core issues in the thesis. It's far less structured than a presentation, and is, while not hostile, definitely critical. I got asked several questions with the intent to stump me, including one that was intended as a lesson for the future.