What if you do it on you desktop?

Server use is typically its own license category. At the company I worked for, their server licensing was based on number of server CPUs (technically only those actually used by the service using the font) and was valid for a year meaning you’d have to re-up annually.

I don't understand why your comment is downvoted because it's absolutely correct (for most type foundries anyway).

The law sees in color, not black and white. If your beige tower desktop computer is talking to users on the Internet, a court can decide you should have used server licensing.

I read the question as "what if the designer uses designer tools to make a PNG with text instead of HTML that calls for a fancy font, then posts that?" It seems like that should (in the US) work like any normal use of a font on a designer's pc^Wmac, and the image doesn't get copyright-like encumbrances from the font.

Exactly. I should have been more clear.

This is the right principle to have absorbed from the article linked, but applied incorrectly.

For either of the hypotheticals here, it would be possible to operate in the way described, subject to some constraints. There is no reason to think that the courts would perceive any legal imperative to recognize the distinction between server licenses or otherwise on copyright grounds. Companies can't invent law and say that it applies just because they wrote down how they'd like for things to work.

The technical reasons referenced earlier are what make these hypotheticals untenable, not law.