Lee's Riemannian Manifolds: An Introduction to Curvature states the following on page 14:
We let $\Lambda ^k (V)$ denote the space of covariant alternating $k$-tensors on $V$, also called $k$-covectors or (exterior) $k$-forms.
This confused me because, for me, a $k$-form is an alternating covariant tensor field of degree $k$, not just a tensor.
Later, Lee defines the following:
the bundle of $k$-forms is $\Lambda^k M := \coprod_{p\in M}\Lambda^k(T_p M).$
a differential $k$-form is a smooth section of $\Lambda^k M$.
According to this, am I correct in assuming the following?
A $k$-form without "differential" is just an alternating multilinear map from a vector space to the space of scalars.
A differential $k$-form is what I have been thinking a $k$-form is, that is, an alternating covariant tensor field of degree $k$.
Some people omit "differential" for some reason when they actually mean differential $k$-form.
If I am correct, is it a common naming method?
Yes, you are correct assuming what you enumerated. It is very common, from my experience.
Note also that, for some authors, a differential k-form is just a section of $\Lambda^k M$, whereas a smooth/differentiable differential k-form is a smooth/differentiable section of $\Lambda^k M$. For practical reasons, once the reader gets comfortable with the naming, all the preceding adjectives are usually ommited for the sake of clarity or brevity and one just says k-form.