In looking into questions of synonymy (or its absence) it is important to keep in mind that individual words and phrases themselves do not have atomic meaning: they have not only ranges of meanings, and different meanings in different contexts, but also different ranges of meanings in different contexts. So even in such distinct contexts as you provide you may discover instances where the two phrases are essentially synonymous, instances where only one or the other phrases is usable, and instances where both may be used but they have different meanings. There is not necessarily a 'core meaning' to any word/phrase, or 'core difference' between two words/phrases, which translates predictably from one context to the next.
Let's look at the phrases 1) by Xself and 2) on X own in three different contexts: a) with do, b) with live, and c) with be.
1a: I did it by myself.
2a: I did it on my own.
- Both by myself and on my own may mean ‘without assistance from anybody else’. But by myself may also mean ‘when I was alone’; and on my own may also mean ‘on my own initiative, without being prompted or ordered to do so’.
1b: He lives by himself.
2b: He lives on his own.
- Both by himself and on his own here may mean simply ‘without any
cohabitant’. But by himself may be used with an implication of
isolation or loneliness, while on his own may be used with an
implication of independence and self-sufficiency.
1c: We're by ourselves.
2c: We're on our own.
- Both by ourselves and on our own be used to mean ‘without
assistance or support’, ‘left to our own devices’; but on our own
may also express a sense of opportunity to exercise initiative. And
by ourselves is more often used to signify privacy—“Alone at last!”—and consequent freedom to pursue more intimate or
surreptitious activities.
[verb] one's way {{in | into | to } [target] | through [hindrance/obstacle]}
is a general pattern where you can substitute different verbs, like "force", "claw", "push", "work", "beg", or really anything that is clearly understandable in the context.
It means, roughly, "Attaining [target] by [verb]-ing, thereby creating a path or entry into [target] (or through [obstacle]) for oneself, where a path or entry wasn't available."
The worm cheerfully munched its way right to the center of the apple. [There was no path to the center of the apple; the worm created one for itself by eating.]
She slept her way into the movie industry. [She had sex with directors or producers in exchange for getting roles in films, thereby creating a way into the industry that was not otherwise available.]
The usage "don't you have to push your way in" is clear: "don't you have to get into the game by means of creating an entry for yourself via pushing?"
"in" is correct, because that's the preposition that is used for indicating participation in a game: being "in the game".
"Push" is also appropriate, because it has figurative meanings, one of which is forcing or asserting oneself. (E.g. an overly assertive person is called "pushy" and "pushes around" other people).
Best Answer