Hey, there, @Shax. I'm a stage actor, and I've had similar experiences when it comes to dealing with nerves. I know that public speaking and acting can be two completely different beasts (in acting, you can "hide" behind being in character; in public speaking, not so much), but I think I might be able to provide some insight.
I have a similar feeling to this when asked to do performance art-related stuff in front of family in private (i.e. this doesn't happen to me when they're in the audience at a paid performance). I don't choke, per se, but the thought of performing in front of them in private just makes me extra nervous and anxious.
I think the reason I feel this way (and perhaps why you do, as well) is because you feel like they're encroaching on your domain by letting them watch you practice. In my life, I tend to be this way with a lot of stuff that I am very guarded about. I have a certain amount of pride when it comes to things that I consider to be my domain (the "I don't need help" bad kind of pride), and that is definitely at the root of why I feel nervous.
I know exactly what you're talking about here. The way I think of it is like this: Imagine if someone gave a good tug on a big lock of your hair. Your head will get yanked, but it won't hurt all that much. Now, imagine if they tugged with the same strength, but only on a few strands of hair. Regardless of how much your head moves, it's going to hurt like the dickens. Same thing with practicing/performing in front of small groups of people. Unlike the whole hair pulling thing, however, I believe that this is something you can get used to over time.
This is actually a very good point of interest: Being memorized and well rehearsed can go a long way in quelling your nerves in any performance situation (i.e. in front of 300 people or 20 people), regardless of whether you can read from the script or not. I'm not sure how much this applies to your particular situation (there are many other factors at play, obviously), but it might have played a small part. I could be wrong, though.
If you have any questions, feel free to ask, and I'll do my best to answer!
Fun Fact: Dustin Hoffman was offered by his community college that he attended for one semester to name their remodelled auditorium after him. He declined (he didn't even go there for acting), and instead requested that there be a bathroom in the backstage area for actors to use, stating that, "when I was in theatre, I always felt like I had to crap a bomb before I went out on stage." So somewhere in the USA, know that there is a Dustin Hoffman Bathroom backstage at some community college auditorium, specifically made for nervous actors.