balk is the American spelling of baulk, meaning to hesitate, stop short, or refuse to proceed due to reluctance, fear, or an obstacle.
"He got personal visits from both Ballmer and Gates that pressured him to stay with Microsoft but did not balk" can be rephrased as "He was pressured by Microsoft [to stay with Windows] but that pressure did not cause him to hesitate [with switching to Linux]."
Merriam-Webster has balk as to stop short and refuse to proceed. To understand the intended meaning, put the emphasis on the first part: If the mayor did not balk, he did not stop, that is, he refused to comply and proceeded with his plan.
> Merriam-Webster has balk as to stop short and refuse to proceed. To understand the intended meaning, put the emphasis on the first part: If the mayor did not balk, he did not stop, that is, he refused to comply and proceeded with his plan.
I feel like one of us is extremely confused.
"Balk" = "stop" = "refuse to proceed", right?
So "<did not> balk" = "<did not> stop" = "<did not> refuse to proceed", right?
"<Did not> refuse to proceed" with Microsoft's proposal therefore means he in fact complied with Microsoft's proposal and did proceed with it, right?
No. Balk = stop and return on your steps. He was moving towards ”migrating more than 14000 PCs in public administration to Linux”, Gates and Ballmer tried to make him stop, but he continued on his path.
In this case, willing to proceed with what he was planning to do, that is, moving from Windows to Linux against Microsoft's wishes.
edit: And you're right that this implies "balking at Microsoft's proposal". Personally, I did read the sentence as intended ("they wanted him to balk (= stop his plans), but he did not"), but I can see how the phrasing can be misleading...