Looking at the etymology in Faldage's link for burl, I guess you're both right. I thought perhaps 'birl' had come from 'burl' in some way perhaps to do with gripping the knots in the log with your feet in order to spin it, but pure imagination I'm afraid! The best I could get for 'birl' was in MW, suggesting it might be imitative, not sure of what. But it has a longish pedigree. If I remember, I'll look in the OED tonight.

Main Entry: 2birl
Function: verb
Etymology: perhaps imitative
Date: 1790
transitive senses
1 : SPIN
2 : to cause (a floating log) to rotate by treading
intransitive senses : to progress by whirling