Indeed, and if the GP has said "there's no downtime in cooking", it would have been apt.
"Cooking onions is not downtime" isn't an effective rebuttal of the general point of "I chop things during downtime, for example when cooking onions".
"Cooking onions is not downtime" isn't an effective rebuttal of the general point of "I chop things during downtime, for example when cooking onions".
"I only tell the truth, for example when lying" is irrecoverably self-invalidating. There is no second example that could nullify its effects. The argument is made invalid by that example specifically. (Not to mention, it calls into question the claimant's understanding of what it means to tell the truth.)
You have misidentified the fallacy here.