The line between definition and simple-metaphor is not always black-and-white. I have broken this down into some specific questions/answers that might help you.
Is it perfectly understandable? Yes. Even if one isn't a walking dictionary, one can consider the metaphorical sense of it. The line between definition and simple-metaphor is be a spectrum. (See next point.)
Are 2nd, 3rd, 4th definitions of some words based on metaphor? Yes. The line between definition and word-as-metaphor is ultimately a popularity contest. The first metaphorical usage of a word (or any other non-standard usage for that matter) is just that: metaphorical. If it becomes ubiquitous, then it's considered part of the standard definition.
Is it an idiom? No. Google search shows this is used in many ways, mostly physical: "at the poke of"
. But if the author is famous, this line becomes famous, and everyone starts using it like that, it (theoretically, but not likely) could become an idiom!
Why "poke"? I might conjecture the author's use of "poke" was to express a personal/emotional link to Obama. The author feels moved by Obama. The point made in the speech was not just a point; it was a call to action. The author considers Obama his/her mentor. A mentor may poke someone to move in a positive direction, and it's in this sense that this particular meaning of "poke" is more quickly understood.
Is this the best prose in the world? Absolutely not. But then, it doesn't have to be; I don't know the context. In informal spoken English, it's acceptable. If it's an answer to a live interview question or an unscripted speech by the person to an audience, the stress to say something quick is even higher, and the person may settle for "poke" which is close to point and prod. "At the ummm (what is that word __blank__ of Obama's speech... point/prod/poke? prod of Obama's speech? no that's not right... point of Obama's speech? No.. it moved me... quick I have to say something!) poke of Obama's speech..."
What would be better? That's a matter of style of course. If this quote is lifted from a famous artist's best selling book, then who cares? But for fun, let's "play editor/critic". The use of "poke" is ironically informal in talking about inspiration to pursue higher education based on a presidential speech. Something like the following might be better:
- Original: "I felt I had to get a college degree at the poke of Obama's speech about higher education being essential in this globalised economy."
- I felt a strong prod by Obama to get a college degree when he talked about higher education being essential in this globalised economy.
- I felt personally compelled by Obama to persue a college degree when he talked about higher education being essential in this globalised economy.
- When he talked about higher education being essential in this globalised economy, I felt personally compelled by Obama to persue a college degree. (Note how putting the long subordinate clause first makes this sentence easier to read than the prior sentence. But it also delays the reference of "he"; that may or may not be a problem, but this would be nicely resolved if it were previously understood to be about Obama.)
Note: I consider this an addendum/addition/complimentary to @BobRodes' excellent answer without which this post would not exist.