In "An Introduction to Modern Astrophysics" Carroll and Ostlie describe the curvature of space by mass as:

curving in a fourth spatial dimension perpendicular to the usual three of "flat space."

They then add in a footnote:

that this fourth spatial dimension has nothing at all to do with the role played by time as a fourth nonspatial coordinate [their emphasis, but it still doesn't clarify things for me] in the theory of relativity.

However in the spacetime wikipedia it says:

Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. ...Minkowski spacetime is flat, takes no account of gravity... The presence of gravity greatly complicates the description of spacetime. In general relativity... spacetime curves in the presence of matter.

And it also mentions the curvature of time and not space:

Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime... [and] says nothing about curvature of the space component of spacetime.

My confusion seems to be about:

• Popular explanations separate the four dimensions of spacetime into 3 space and 1 time, so how does curvature in a fourth spatial dimension not lead to understanding spacetime as being 4 spatial dimensions and 1 time dimension?
• How is a curvature of 3D space interpreted as curving in a fourth spatial dimension while having "nothing at all to do with the role played by time as a fourth nonspatial coordinate"?