The word "vector" just means an element of a vector space: a collection of "things" that you can add together and multiply by numbers. The vector space R^n is defined as the collection of all lists of n real numbers, so you are right in that sense.
But usually when people say that a vector "is" a list of numbers, they mean that if you choose a coordinate system, then the coordinates of a vector uniquely identify it. This is exactly analogous to locations on the earth being uniquely identified by their latitude and longitude: we first have to make a (somewhat arbitrary) choice of coordinates. You wouldn't say that a point on the earth "is" a pair of numbers, though. The case of R^n is confusing, because if you choose the usual coordinate system, then the coordinates of a vector are the same list of numbers as the vector itself.
I like to think of tensors as functions that map some number of vectors to some other number of vectors. But you can add together two tensors (with the same number of inputs and outputs) or multiply a tensor by a number, so you can also think of them as vectors in some other vector space.
But usually when people say that a vector "is" a list of numbers, they mean that if you choose a coordinate system, then the coordinates of a vector uniquely identify it. This is exactly analogous to locations on the earth being uniquely identified by their latitude and longitude: we first have to make a (somewhat arbitrary) choice of coordinates. You wouldn't say that a point on the earth "is" a pair of numbers, though. The case of R^n is confusing, because if you choose the usual coordinate system, then the coordinates of a vector are the same list of numbers as the vector itself.
I like to think of tensors as functions that map some number of vectors to some other number of vectors. But you can add together two tensors (with the same number of inputs and outputs) or multiply a tensor by a number, so you can also think of them as vectors in some other vector space.