As has been mentioned in the comments, although you have gathered together quite a few definitions and examples of usage, a common thread runs through them all in terms of what 'figure out' means.
In a broad sense of the word, to figure out something means to understand something clearly, however, what differentiates it from similar expressions such as to realise something or to find out something is the process involved in coming to that conclusion.
When we realise something or find out something, we are able to comprehend something that we couldn't comprehend before, but that may not have been our intention. Realisations can often happen by chance - something may spark a connection, and you then see it in a new light.
For example:
I just realised that there's an arrow hidden between the E and the
x in the 'FedEx' logo!
Similarly with 'finding out something' - we may have come by the information passively.
I just found out that my girlfriend was cheating on me.
Figuring something out, on the other hand, implies a conscious effort to study a matter or situation and actively search for the answer or understanding of it. Rather than by chance or happenstance, it is normally done with a specific goal in mind.
For example:
- We need to figure out what went wrong.
- It took them ages to figure out how to assemble the chest of drawers.
- When you've figured the answer out, put your hand up.
- They spent 20 minutes figuring out how to open the box.
There is a broad range of examples above, but all of these exhibit intent to come to a conclusion.
Best Answer
The first sentence doesn't really feel like natural English, but I guess if you pressed me for an answer, I'd say that the sentences are syntactically the same.
To me the proper idiom is to "push (someone) into (doing something)" since "push to" is not a phrase commonly used for people. What I mean is, I can "push my car to the gas station" but I don't usually "push my friend to the pub". I can carry him to the pub, or walk him to the pub, or force him to the pub, or even drag him to the pub, but not push.
Don't ask me why. Idioms don't always make sense. Also "push to" might be fine in other English-speaking regions than Southern California.