Frying onions is not downtime.

This is the "your example is invalid, therefore your argument is invalid" fallacy.

It's not a fallacy. There is downtime in baking, but not cooking.

"There's no downtime in cooking" is different from "cooking onions is not downtime". Though you'd still need to clarify "even if stuff is already chopped", I guess.

“Cooking onions is not down time” directly follows from “there’s no downtime in cooking.” If there is no downtime in cooking is true, then *cooking* onions cannot be downtime.

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".

If cooking onions is not downtime, then it absolutely is an effective rebuttal to the general point of "I chop things [only] 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.

How is "there's no downtime in cooking" better than "Cooking onions is not downtime?" Both are assertions without backing evidence. For instance, it's possible for "Cooking onions is not downtime?" to be true (e.g. cooking onions requires active work) while "there's no downtime in cooking" is not true (e.g. cooking chicken broth involves downtime).

Because one is an effective counterargument if true, the other isn't, even if true. Hence, the former is a better counterargument.

It kind of is, versus an active task like chopping.