I thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.

> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).