Wikipedia defines a surface to be a two-dimensional manifold, and a closed surface to be a surface that is compact and without boundary. Am I correct that this definition of "closed surface" is not equivalent to the definition "a surface that is closed (as a Hausdorff space)"? A "closed surface" is defined to be compact, but a compact Hausdorff space is closed.
So, for example, a disk that includes its boundary is not a closed surface, but it is a surface that is closed (topologically). I find this definition very confusing.
Edit: As Joppy points out in their answer, this same issue applies to any closed manifold, not just two-dimensional surfaces.
You're partly correct. A "closed manifold" is a manifold which is compact and has no boundary, and a "closed surface" is just a specialisation of that definition to dimension 2, and this is not equivalent to the definition of being closed as a Hausdorff space.
But any space is closed in itself, so every surface is topologically closed, so this is an uninteresting fact. This is why the re-use of the word "closed" is not too confusing.