"negative x" is "that which negates x [via addition]", which makes sense no matter what the nature of "x".
As for "positive" ... I don't know exactly what the "New Math" designers were thinking, but my sense is that the word is essentially a pre-emptive reaction to introduction of the term "negative". "Positive 3" is just the 3 you've known since you started counting, the 3 that describes how many elephants can be put here ("posit" = "put"), the "regular" 3, but with a fancy-sounding name applied in explicit contrast to "negative 3", the new-fangled number that cancels-out that familiar one.
When I'm near a mirror, there's (left-handed) "me" and (right-handed) "reflected-me". If I want to discuss the two of us, I might occasionally need to emphasize that the lefty is me --this one, here!-- so I might be inclined to use a phrase like "actual-me" or "original-me". In this sense, "reflected" is like "negative", and "actual" is like "positive" ... a word I apply merely help make a distinction ... an explicit redundancy, given my natural bias toward half of the "people" near the mirror.
Tying the analogy back to my original point: Given that "reflected-(reflected-me)" is (barring existence of additional mirrors) "original-me", we have that "reflected-x" makes sense regardless of the nature of "x". Thus, "reflected" --as "negative"-- describes an object's relationship to a counter-object, not its relationship to an intermediate object (the mirror, or zero).
That we refer to numbers less than zero as "the Negatives" is just a (ahem) reflection of our bias toward "the Positives", which seem --based on our foundational counting experience-- more like "actual" quantities.
I only use "minus" to describe the subtraction operation in arithmetic.
Edit to add:
Actually, I also use "minus" to describe a number less than zero: literally, a number that, via addition, makes a quantity smaller. (Latin "minus" = "less") Compare "plus": a number that, via addition, makes a quantity larger. (Latin "plus" = "more") Being a "plus" or a "minus" is an inherent property of the number's value, describing its relation to the intermediary, zero.
That said, I admit that I'm often a bit loose with my terminology. I might describe a number less than zero as "negative"; but, again, that's part of our bias toward what we call the positives.