Yes Phillip.....ditto what Mum said.
My interest in psychology led me to grasp a few basics, one of which is.....
Think

Feel

Do.
What we think....greatly effects what we feel....and eventually what we do.
So..someone pushing our buttons...causes us to think something about that....which then induces a feeling....and might then see us act. (to use your example).
Depending......as you say, on our decisions inbetween (our choices)...we will act accordingly.
Thus..in theory...if we choose to think:
"Oh golly gee that goof is bugging me" (after the button-pusher begins to push)...
we may very well feel......darn ticked off.....and may also, as you say, react and do...something in response, or choose to act a certain way.
And....to carry that further......as you point out...we have further choices....we could decide, after that first thought:
"Naw....phooey on that dork! Not letting him/her/it wreck my day!"...
so then we may very well feel.....not ticked off at all....and may also do something else, after that, possibly something to be proud of.
The big secret in it all, I think, is.....catching oneself thinking stuff and then...loving oneself (forgiving oneself for thinking whatever unkind thoughts, or doing whatever undesirable actions, or having whatever not so wonderful reactions......to the button-pusher's behaviour), in the first place. We can then make better choices. And ultimately....developing the skill to catch oneself...prior to action of any type...and considering all options....making better choices first...is the objective.
It's great stuff and it is a skill I believe we call all learn to apply....at least some of the time.
I was quite surprised, not that long ago, when I took a parenting course...just to see what is being taught these days....to try to hone my skills (so to speak)...and to sort of make sure I was doing the best I could and not missing important new ideas.
The first thing the instructor wrote on the board was:
Think

Feel

Do.
Same stuff. Different time. I was a bit shocked but glad to see that what I had embraced, over all of those years, still makes sense and is being taught today for the benefit of these parents and their children. It was a good experience and I learned some new things too. And my belief in the good old basics was reinforced.
GFN