What is a differential form?

I have encountered several different objects that people call with the same name “differential form” and I would like for more experienced people to help me clear the ambiguity and tell me which of the following is the true/most used/most useful definition of differential form. For simplicity, I will only talk about differential 1-forms, hopefully the answer will automatically generalize to k-forms. I’m assuming a smooth n-dimensional manifold M with no extra structure. Maybe as an extra, if you know, you can also tell me what the other listed objects that are not differential forms are called! One A differential form ω is a smooth section of the cotangent bundle. That is, ω : M -> T^{star} M with ω(p) = ω_p, where ω_p lives in T^{star}_p M, meaning that ω_p takes as input a tangent vector at p and outputs a real number. Two A differential form ω is a map ω : TM -> R, with ω(p, v) = ω_p(v) where ω_p is the same as above, TM is the tangent bundle and v lives in T_p M. Three A differential form ω is a map from (set of all sections of TM) to C^{infinity}(M) , with ω(X) = f where X is a vector field and f a smooth function on M, and we evaluate f by f(p) = ω_p(X(p)) where ω_p is the same as before and X(p) = v is a vector in T_p M. Four A differential form ω IS the linear functional ω_p we’ve been talking about up until now, meaning that a differential form only makes sense after you’ve specified a base point. To summarize: what does a differential form take as input? A point in M, a pair consisting of a point in M and a tangent vector based at it, an entire vector field, or a tangent vector at some point p? Thanks to anyone who answers!