This is a tremendously common confusion to have, and in my experience, people are notoriously bad at explaining this concept. I'm sorry that you had to deal with people who were abrasive in addition to poor expositors.
In an arbitrary vector space, you cannot talk about components. They actually don't exist. Now, you can impose them on a finite-dimensional space by providing a bijective linear transformation from the arbitrary vector space to $F^n$, but then they're just that: an imposition, because any other bijective linear transformation will choose different would-be "components".
Components exist in $F^n$ because of the actual nature of the objects involved. So you don't need a basis, you can just look at an arbitrary object $(a,b,\dots,n)$, and find any of its components, because they're built into the object. This can be confusing because we also write coordinate vectors in this way, and when the basis is the standard basis, there is no difference between the components and the coordinates. However, in any other basis, there will be a difference.
(Edit: Val made an important point in the comments. I should have been more careful when I said there was "no difference". The fact is that coordinates and components are never conceptually the same, but I meant to say that in the standard basis case they will be numerically equal.)
Lacking a basis at all, you might want to say that $F^n$ still has coordinates implied by its components. But, in my opinion, this seems silly, since you cannot do the same in other spaces.
So the short answer is: Yes, there is a difference, because components are part of the objects.
As for your "collection of vectors" notion, they are basically the same. But it is easy to imagine a collection of vectors which is not a vector space: for example the circle in $\mathbb{R}^2$. This is definitely a collection, and the objects in it are definitely vectors, but it is not a vector space.
What I assume you meant by "collection" was what we might call a "meaningfully structured collection", and the meaningful structure is described precisely as an abelian group over which elements can be scaled by objects in a field. In that sense, your notion is correct, though a bit less transparent.
The "formal Pythagorean distance" is not invariant under change of coordinates, i.e., it depends on the choice of coordinate system (e.g., multiply Cartesian coordinates by a positive constant, or consider polar coordinates).
The Riemannian-geometric way to interpret the Pythagorean distance is via arc length with respect to a metric (smooth field of inner products) $g$: If $\gamma$ is a piecewise $C^{1}$ path satisfying $\gamma(a) = p$ and $\gamma(b) = q$, the length of $\gamma$ over $[a, b]$ is
$$
\ell(\gamma) = \int_{a}^{b} \sqrt{g(\dot\gamma, \dot\gamma)};
$$
the distance from $p$ to $q$ is the infimum of $\ell(\gamma)$ over all such paths joining $p$ to $q$.
The Pythagorean formula comes from the constant field $g$ whose value at each point is the Euclidean inner product, with components $g_{\mu\nu} = \delta_{\mu\nu}$; the shortest paths are easily shown to be line segments, and the length of a line segment is given by the Pythagorean distance.
For a general metric $g$, however, the distance is not given by the formal Pythagorean distance.
Best Answer
Coordinate transform is a technical term. It refers to the process of finding out the new coordinates of a point fixed in space when the coordinate system is changed. "Change of coordinates" is not really a technical term. When a point $P$ has its coordinates changes from $(x_1,y_1)$ to $(x_2,y_2)$, it could be that the point is physically moved in the space or we are simply moving the coordinate system while the point is fixed is space. Both actions will cause change of point $P$'s coordinates.
If your book is making a clear distinction between "coordinate transformation" and "change of coordinates", I think it is using "coordinate transformation" to refer to the action of moving the coordinate system and using "change of coordinates" for moving the object. These two are indeed distinct concepts.