In coordinate-free relativity, how do we define a vector?
Honestly, this coordinate-free GR stuff (Winitzki's pdf in particular) looks like GR as would be taught by a mathematician--very similar to do Carmo's text on Riemannian geometry. In classic (pseudo-)Riemannian geometry, vectors are defined as derivatives of affine parameterized curves, covectors as either maps on vectors to scalars or as gradients of scalar fields. Something like the Riemann tensor is defined as a map on two/three/four vectors spitting out two vectors/one vector/a scalar.
Differential geometers love defining everything as a mapping; I consider it almost a fetish, honestly. But it is handy: defining higher-ranked tensors as mappings of vectors means that the tensor inherits the transformation laws of each argument, and as such, once you establish the transformation law for a vector, higher-ranked tensors' transformation laws automatically follow.
Edit: I see the question is more how one can figure out a given physical quantity is a vector or higher-ranked tensor. I think the answer there is to look at the quantity's behavior under a change of coordinate chart.
But Muphrid, we never chose a coordinate chart in the first place; isn't that how coordinate-free GR works?
Yes, but the point of coordinate-free GR is just to delay the choice of the chart as long as possible. There is still a chart, and most results depend on there being a chart, just not on what exactly that chart is.
How does looking at a change of chart (when we never chose a chart in the first place) help us?
The transition map from one chart to another is a diffeomorphism, and so its differential can be used to push vectors forward or pull covectors back. Hence, the transformation laws that usually characterize vectors and covectors are still there. They look like this: let $p \in M$ be a point in our general relativistic manifold. Let $\phi_1: M \to \mathbb R^4$ be a chart, and let $\phi_2 : M \to \mathbb R^4$ be another chart. Then there is a transition map $f : \mathbb R^4 \to \mathbb R^4$ such that $f = \phi_2 \circ \phi_1^{-1}$ that changes between the coordinate charts.
Thus, if there is a vector $v \in T_p M$, there is a corresponding vector $v_1 = d\phi_1(v)_p \in \mathbb R^4$ that is the mapping of the original vector into the $\phi_1$ coordinate chart. We can then move $v_1 \to v_2$ by the (edit: differential of the) transition map.
But Muphrid, aren't we meant to be working with the actual vector $v$ in the tangent space of $M$ at $p$, not its expression in a chart, $d\phi_1(v)$?
You might think so, but (as was drilled to me repeatedly in a differential geometry course) we don't actually know how to do any calculus in anything other than $\mathbb R^n$. So I think there's some sleight of hand going on where "really" what we do all the time is use some chart to move into $\mathbb R^4$ and do the calculus that we need to do.
What this means is that, in my opinion, coordinate-free is a bit of a misnomer. There are still coordinate charts all over the place. We just leave them undetermined as long as possible. All the transformation laws that characterize vectors and covectors and other ranks of tensors are still there and still let you determine whether an object is one or the other, because you're always in some chart, and you can always switch between charts.
There are 4 common definitions of tangent vectors, some of which make use of coordinates only casually or even not at all.
Definition via transformation laws
There's a somewhat technical one preferred by some physicists (those who value calculation rules over geometric insight - shut up and calculate, you probably know the type): A vector is just an $\mathbb R$-tuple that obeys certain transformation laws under a change of coordinates.
This definition actually makes sense in context of the Erlangen program, as the tangent space is a vector bundle associated to the principal bundle of linear frames. However, as tangent spaces are normally introduced long before lie groups and principal bundles, the definition appears unintuitive.
Definition as equivalence classes of curves
A more intuitive one defines a vector as an equivalence class of curves tangent to each other. We need to make use of coordinates to define the necesary first-order contact of curves, but this use is far less prominent.
This definition makes clear why tangent vectors should be considered velocities and comes with a natural generalization to higher jet spaces.
Definition as derivations
We arrive at a totally coordinate independent characterization by identifying vectors with their directional derivatives: A vector is just a derivation, ie a linear functional that respects the Leibniz rule.
That's the definition that can be found in (most?) modern literature on differential geometry (where modern means something like the 60s).
Algebraic definition
Another coordinate-free (but very abstract one) comes from algebraic geometry, and Muphrid braught it to my attention just yesterday: There's a purely algebraic definition of the cotangent space, and the tangent space is just its dual.
I suspect the algebraic definition can probably be made more concrete (from an analytic point of view) in terms of infinitesimals (see Advanced Calculus by Sternberg for a definition of infinitesimals that makes sense in standard analysis, but is of course not identical to the non-standard one).