I read various definitions for a tensor (and watched introduction videos), but still have a small confusion, where I kindly ask for clarification.
One definition that I really like and found often is:
A $(n,m)$-tensor is a linear scalar function (an operator) that takes $n$ covectors and $m$ vectors in input to return a real number (scalar) as a result. It is covariant in $n$ components and contravariant in $m$ components (which is still clear for me).
And now my confusion starts when reading the additional fact (see for example this post, this post or this paper):
We say that a tensor has "type $(n,m)$” when it is of contravariant order $n$ and covariant order $m$.
Is there a simple explanation for this or I am just mixing $m$ with $n$?
In this context I ran into a related confusion: Not only in this paper it is stated "Tensors of type $(1,0)$ are vectors. Tensors of type $(0, 1)$ are covectors". Contrary, in Johnston's book "Advanced Linear and Matrix Algebra" it is stated $(1,0)$-tensors are row vectors and $(0,1)$-tensors are column vectors. Usually row vectors are in the literature, as for example in this post, treated as covectors and column vectors are treated as (contravariant/tangent) vectors. There is even a Mnemonic given by this post: "covariant = row vector; because COvariant sounds like ROWvariant."