As a layman, the word "tensor" always intimidated me. As a programmer, I was surprised then when I found out that a tensor is just a multi-dimensional array (where the number of dimensions can be as small as 0). That was a concept I was already quite comfortable with.
That's a bit like saying a vector is 'a row of numbers'. Not incorrect, but missing the point. What matters is what vectors do. It's the properties like pointwise addition, scalar multiplication, and existence of an inverse that make vectors vectors.
You're confusing a tensor with its representation. Tensors are objects which must obey a certain set of rules. (Which rules depends on whether you're talking to a mathematician or a physicist.)
