Difference between Cross-Cap, Mobius Band, and Real Projective Plane

I understand how the mobius band and the real projective plane are different (the first is a manifold with boundary, for example, while the second is a compact, closed surface and a true manifold).

1. Is a cross-cap the name for spaces which are homeomorphic to the mobius band and which are sewn into holes cut from surfaces in order to increase their non-orientable genus? (The orientable counterpart being handle attachment -- the mobius band, in contrast, has a "twist".)

2. Or is the cross-cap what results from the connected sum of a disk and a mobius band, i.e. the real projective plane?

Core Question:

I.e. (up to homeomorphism) is a cross-cap 1. a mobius band or 2. a real projective plane? Or something else entirely?

The terminology does not seem to be consistent between sources, for instance Wikipedia https://en.wikipedia.org/wiki/Cross-cap, or the various mentions of the term "cross-cap" in Gross, Tucker, Topological Graph Theory.