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.