What is the clear difference among these synonyms of 'push'? Are there any default associations in the speaker's mind with these verbs?
You can twitch the rug straight though. For me:I don't think that jerk is a synonym of push either. To jerk something is to move it violently, and that violent movement could be a pull or a push.
I didn't think that twitch was a synonym of push, but my dictionary may disagree. It says that twitch can be a transitive verb, but I've only ever heard it used as an intransitive verb. That is to say, I can twitch, and I can make you twitch using some stimulus, but I can't twitch you.
I'm not sure what I would say about shove. It's a synonym of push, but I don't know what you mean by default associations.