Why does this magically change when someone utters the words “the internet?”
You are not doing business in another country. First, there may not be any "business" involved at all. More importantly, however, you are not in the other country. The other person is. If physical property is involved then someone—not necessarily you—is going to need to worry about import/export regulations when moving it across the border. When it comes to virtual services and websites, however, those concerns rooted in physical transportation do not apply.
> Why does this magically change when someone utters the words “the internet?”
It makes no difference whether you're communicating over the Internet or by phone or two-way radio or postal mail or carrier pigeon, or by shouting to each other across the border. So long as you're only exchanging data and not physical goods they can implement technical measures to block the communication at the border, or order their own residents not to communicate with you—in which case I would advise them to make arrangements leave the country posthaste—but you yourself remain outside their jurisdiction.
This is something that’s already heavily regulated for other industries e.g. financial services. There’s nothing novel about the GDPR in that aspect.
Services which involve nothing but data, and thus still fall under the category of "data moving over borders". And, consequently, freedom of speech.
May I introduce you to services such as Wikipedia.com and news.ycombinator.com?
Which is literally just data.
> a forum through you can interact with other people
Which is just a means of exchanging data.
What is missing in both cases is any transportation of people or material across the border.
Regardless of whether or not your model actually makes sense, an inevitable consequence of your model is that the internet will become siloed at a national level. It sounds like you’re fine with that though.
You're not in their "house". You're just communicating with someone who is. If some foreign government has a problem with that communication they should take it up with the party who is physically within their jurisdiction—not that I believe they have any just standing there either.