The nonjudgmental term of art appears to be "adposition". It isn't clear to me why this was coined as "adposition" and not "apposition".
"Preposition" is already something of an unnatural word class. They have two main functions in English: to describe some kind of location ("Sam? He's with Natasha"), and to mark certain arguments to verbs ("The tools? Sam's playing with them").
It's just a coincidence that prepositions serve both of these purposes in English; in Mandarin, the same two purposes are served by completely separate word classes. Verb argument markers are preposed; location markers are postposed.
> It's just a coincidence that prepositions serve both of these purposes in English
Not really… these categories fall together in a lot of languages. It doesn’t even require prepositions: a lot of languages have case markers which can be used for both location and verbal arguments.
The specific examples you’ve given are, respectively, called the ‘comitative’ and ‘instrumental’ uses of with. These two categories are particularly prone to falling together, mostly because the boundaries between them aren’t particularly clear: for instance, if you say, I made the coffee with milk, you could say that the coffee is accompanied by the milk (comitative), but you could also say that you’re using the milk (instrumental).
"Preposition" is already something of an unnatural word class. They have two main functions in English: to describe some kind of location ("Sam? He's with Natasha"), and to mark certain arguments to verbs ("The tools? Sam's playing with them").
It's just a coincidence that prepositions serve both of these purposes in English; in Mandarin, the same two purposes are served by completely separate word classes. Verb argument markers are preposed; location markers are postposed.