What is the semantic between the terms "take someone on a vacation" and "take someone for a vacation"?

Yes to this. A Practical English Grammar, by A.J. Thomson and A.V. Martinet indicates several verbs that are used this way, including come and go:

“You’ll just have to go/come and see for yourself.”

“On” and “for” have much more potential difference in meaning than “in” and “on”. “For” implies a purpose more clearly than “on”. But people rarely consider the differences to matter at all. An author trying to find the wording that best fits the context might consider the differences, I don’t think people consciously notice these details.

You may well be right, but as a Brit who has lived in the US for a long time, the “visit with” phrase seems to be much more used here than just plain “visit”?

Oh well, language is defined by usage, not dictionaries. The apostrope wars are probably already lost, and ‘alot’ is apparently a word now…