But to confirm, the particular quote from the BMJV that I was going by on that page is this one, does it match with your understanding? (My German comprehension isn't great.)
"Die Anbieterkennzeichnungspflicht muss praktisch von jedem, der ein Online-Angebot bereithält, erfüllt werden. Etwas anderes gilt nur bei Angeboten, die ausschließlich privaten oder familiären Zwecken dienen und die keine Auswirkung auf den Markt haben. Im Zweifel sollten Sie davon ausgehen, dass die Anbieterkennzeichnungspflicht besteht."
The first sentence seemed to say that "in practice everyone offering something online must comply", and I took "privaten" in the second sentence to imply private as in "password protected / not publicly accessible". But as someone learning German I've probably interpreted those too narrowly, and I'd love to be corrected on the nuance!
which exclusively cater to private/family needs/affairs and [thus] have no impact on the markets.
> I took "privaten" in the second sentence to imply private as in "password protected / not publicly accessible".
No, privaten Zwecken here refers to personal affairs.
Geschäftsmäßig doesn't require Gewinnerzielungsabsicht (intent to make a profit), but does actually require an intent to make business, in whatever form, but not necessarily on the site itself. For example, a business owner having a very simple website just stating what his business does will require an imprint on that site, because obviously the intent of the website is to inform about his business. If, on the other hand, you have a blog with no ads about your kittens then you don't need an imprint.
Just because something is an unbestimmter Rechtsbegriff (non-exhaustively defined law term) doesn't mean you can just go ahead and act as if it means anything you like.