Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.

That's exactly correct. It's similar to how a vector in R^2 is just an arrow with a magnitude and a direction. When you represent that arrow in different bases, the arrow itself isn't changing, just the list of numbers you use to represent them. Likewise, tensors do not change when you change bases, but their representations as n dimensional grids of numbers do change.