Oh, I agree. I should have been clearer.
I don't really consider what you're describing as a shove. More like a push to follow up with whatever technique you like, I've used it a lot with hook punches. Both real close in hooks and slightly wider ones.
I've used that little push with a roundhouse, too, bringing it angled up over their shoulder. Kind of like what we called a "shovel hook" punch, which is a part uppercut, part hook punch, only with a roundhouse kick - it has a little angle of a front kick.
I always considered a "shove" as moving someone right off his feet, off his stance, with quick power blast, like you would use in a weight lifting snatch....
With a shove, at least the way we use it, it's a full body, leg driven double hand ooomph meant to move somebody. If they're taller and bigger you angle them slightly upwards, raising their center of gravity a bit. Obviously, you have to be in the proper position to do so.
Sometimes in crowds or riots, the "shovee" crashes into others and clears more of a path. I know, I know, that's a politically incorrect thing to say these days, especially as a cop, but I've had to do it in that past. Worked beautifully and nobody was really hurt. If you've never been in a riot, they're probably just like you imagine, they suck. And there's a big difference between a gathering, group protest and a full blown riot.
As a self defense technique dealing with one or a few, a shove really moves somebody several feet, or off their feet completely and down. You can follow it immediately if that person has to be controlled or move to the second person you have to deal with
That's what I mean as a shove. Full body, full power application of your legs, hips, back and arms. With practice, it can be a thing of beauty.