"Pushing" film refers only to a modification of the standard development process.
It has nothing directly to do with exposure.
If you meter using a higher than box speed EI, you are intentionally under-exposing the film.
If you under-expose, you can save a portion of the image by using a push development to increase the contrast of the near-shadow parts of the scene - often at the expense of highlight detail.
Instead of referring to it as "pushing" film, it would be better to refer to it as "intentionally under-expose" film.
I think that you understand this, but may not understand how confusing the nomenclature can be.
EDIT: and as for the question about whether to request that the lab "push" the development, the answer depends on whether you want the contrast increased. "Pushing" won't help (much) with the shadow detail, but it will improve the contrast in the near shadows. So if the scene and lighting were low in contrast, then a "push" would be potentially useful. If the scene and lighting were high in contrast, then you will lose quality in the highlights if the film is "pushed".
If the scene and lighting were high in contrast, only you can decide what is more important - the near-shadow contrast or the highlight detail.
I think it is useful to remember that for most (or possibly all) of the films you list in your first post, Kodak recommends no change in development for film exposed at one stop less than the ISO speed.