Is the term "vector" used exclusively to refer to quantities of two dimensions, or can it also refer to quantities of more than two dimensions, for instance, magnitude, direction, and weight?
Mathematicians don't like to categorically say what the basic things of study are. It has turned out to be much more fruitful to say how things behave, and then every time we come across something which behaves that way, we can apply all the theory we already developed to that thing.
Vectors is one of those concepts that we have defined in terms of behaviour rather than content, and that behaviour is summed up in a few neat axioms. Those axioms basically says that for a collection of "things" to deserve the name vector space (and thus for each "thing" to be deemed a vector):
You can add things from your collection together to make a new thing (and this thing has to be in your collection already). You can scale a thing to make a new thing (and this new thing has to already be in your collection). This adding and scaling work together through distribution (the way regular addition and multiplication work together through distribution).
The actual list is a whole lot more formal, and I glossed over a few requirements, but this is the gist of it. If your collection follows the demands, then it gets to call itself a vector space, and the things become vectors.
So any time you have a collection of things that have this particular vector-space-behaviour, then what you have is a vector space and each thing in that collection is a vector. Dimension is not a part of what we require of a vector space. In particular, a dimension of 2 is not something we require; plenty of commonly used vector spaces (even some in use in middle and high school, without the students realizing it) even have an infinite number of dimensions.