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I've been browsing questions regarding 1-forms and their difference with co-vectors, and I have stumbled upon what follows.

@magma, here, said:

1-forms are simply the linear operators that take a vector and give out a number

So, thinking of a function, the range of a 1-form is a real value.

On the other hand, @Silly Goose, here, said

... if you have a 1-form α on some manifold M and you pick any x∈M, the value of α at x is a co-vector.

Doesn't the latter quote assert that a 1-form, functionally, returns a co-vector, thereby contradicting the former quote?

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    $\begingroup$ A covector is an element $\omega\in T^*_xM$ and therefore defines a linear function $\omega: T_xM\rightarrow \mathbb{R}$. A $1$-form is the dual analog of a vector field. For each $x\in M$, it defines a covector $\omega(x) \in T^*_xM$. It could be called a covector field. $\endgroup$
    – Deane
    Commented Dec 20, 2022 at 14:34

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Remember that you need to put in two inputs into a 1-form before you get a scalar: a point on your manifold, and then a vector at the tangent space of that point. If you fix the point $x\in M$ but not a vector $v\in T_xM$, then you have a covector $\omega|_x:T_xM\to \mathbb{R}$, i.e. $\omega|_x\in T_x^*M$. If you fix a vector field $X$ on $M$ but not a point, then you have a function $\omega(X):M\to \mathbb{R}$. If you fix neither, then you have the usual definition of a 1-form, namely $$\omega:M\times T_x M\to \mathbb{R}.$$

The abstract thing that's happening is if you have a map $X\times Y\to Z$, you can view this same map in three ways. First, exactly as written. Second, if you fix an $x\in X$, this becomes a map $Y\to Z$. Third, if you fix a $y\in Y$, then this becomes a map $X\to Z$.

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These are subtly different. All one forms on a given (assume finite dimensional vector space) are linear maps from vectors to the field. On a manifold each point has a vector space associated with it (the tangent space $T_pM$) upon which one can define one forms as above. The collection of all one forms is commonly denoted as $T_pM^*$. The difference here is that now a one form on a manifold is a function: $$\omega:M \to T_pM^*$$

And the properties must interact in a "nice" way with the smooth structure of $M$ or if you like the tangent bundle $TM$.

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The reconciling difference is in one word that is missing in the latter quote: differential 1-form. Differential 1-form is indeed a smooth field of (algebraic) 1-forms. This is described in @Nick's answer

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