The table of display values defines run-in as

inline box with special box-tree-munging rules

Outer Display Roles for Flow Layout defines it as

Run-in elements act like inlines or blocks, depending on the surrounding elements.

And Run-In Layout says

A run-in box behaves exactly as an inline-level box, except [...]

The first definition seems to imply that run-ins are inline boxes, but with some additional special behavior. The third definition seems to imply that run-ins are not inline boxes, but they behave similarly.

Which is it? For example, if some property says it applies to inline boxes, does it apply to run-ins too?

I think the second definition is just an obsolete copy-paste from CSS2. The run-in never acts as a block (but can generate an anonymous block parent).