Mathematical definitions are just entries in a dictionary, translating between one language and another.

There is certainly power in choosing names and formulating definitions -- I think of it as the "power of Adam". Good names and good definitions will get used a lot, poorer names and poorer definitions won't. There are even aesthetic issues that come into play in deciding between different terminology. For example, one of my personal aesthetic criteria is to avoid acronyms. Also, I know of mathematicians who dislike personal names being attached to mathematical objects, although that's a hard issue to fight against.

Nonetheless, for two different systems of mathematical terminology and definitions, there will be a dictionary that can be used to translate between them. Ideally there will even be a "compiler" that will do that translation automatically and efficiently, just as there are natural language translation devices that convert English to French and back (with admittedly comic outcomes sometimes...)

The translation between two different definitions of "positive and negative" in your question is a simple example of this. As long as the reader knows what "positive" means in the context of what they are reading -- and it is the author's responsibility to be clear on that point -- the reader should be able to make the translation automatically and efficiently into whatever language they are more confortable with.