With family and friends you can just be yourself[1] though. If they watch the recording after 5 or 10 years, they will probably look at it affectionately, at most with some light teasing.

They're not going to, for example, start harassing you online just because at some point during your presentation you said the word "blacklist" (or other words that are nowadays are considered worse than they were back then).

So I'd say part of the reason of why it's easy with family/friends but stressful with strangers, is subconscious fear of this kind of judgement where you don't know how this large amount of people will react, and fear that something you say or something you do will cause a response way more negative than you expect. Sure, you can imagine some possible negative responses, but because you don't know them personally, you don't know what's their "upper bound" for a negative response (how far they will go).

And once you start talking the nerves go away because, once you start, running away suddenly becomes a worse choice (easier to get a negative response) compared to finishing the talk.

That would be my guess.

[1]: Some might cross lines, but those should be the exception, and you probably wouldn't be relaxed with them anyway.