Ray tracing refers to the act of tracing rays. You can use it for lighting, but also sound, visibility checks for enemy AI, etc.
Path tracing is a specific technique where you ray trace multiple bounces to compute lighting.
In recent games, "ray tracing" often means just using ray tracing for direct light shadows instead of shadow maps, raytraced ambient occlusion instead of screenspace AO, or raytraced 1-bounce of specular indirect lighting instead of screenspace reflections. "Path traced" often means raytraced direct lighting + 1-bounce of indirect lighting + a radiance cache to approximate multiple bounces. No game does _actual_ path tracing because it's prohibitively expensive.
I believe the "path tracing" you described here is actual path tracing insofar each sample is one "path" rather than one "ray", where a "path" does at least one bounce, which is equivalent to at least two rays per sample. Though I think the "old" path tracing algorithm was indeed very slow, because it sent out samples in random directions, whereas modern path tracing uses the ReSTIR algorithm, which does something called importance sampling, which is a lot faster.
The other significant part is that path tracing is independent of the number of light sources, which isn't the case for some of the classical ray traced effects you mention ("direct shadows" vs path traced direct lighting).
Path tracing is a specific technique where you ray trace multiple bounces to compute lighting.
In recent games, "ray tracing" often means just using ray tracing for direct light shadows instead of shadow maps, raytraced ambient occlusion instead of screenspace AO, or raytraced 1-bounce of specular indirect lighting instead of screenspace reflections. "Path traced" often means raytraced direct lighting + 1-bounce of indirect lighting + a radiance cache to approximate multiple bounces. No game does _actual_ path tracing because it's prohibitively expensive.