I can understand the confusion. I treat "do your best" as an idiom rather than an imperative. I think it's usually said in one of two contexts: (1) as a platitude similar to "good luck" or (2) as a response to the expectation of a subpar result. In the former, it could be a friend hyping me up before riding in a cycling race. The latter could be me telling my boss that we're going to miss a deadline and they respond with that to tacitly tell me to finish it as soon as possible.

To your second question, it's the latter. In my job, I work with a lot of print and I'm often juggling dozens of print assets at once. I'm always "doing my best" because my livelihood depends on it, but how much effort I put in to each project varies. If it's a small project where we're scraping pennies, I'm more inclined to let small imperfections slide. If it's a major project, I'm nitpicking everything. Obviously, I want everything to be good, but I have to triage my efforts and I'm expected to do so.

To me, "doing my best" isn't about perfection so much as maximizing results with as much effort as I have to give. If putting in x effort gets me 90% of the way to perfection, but 2x effort would get me to 95%, I'm inclined to stop at 90% unless perfection is needed or desired. How I quantify that percentage varies from task to task. If I'm painting a miniature, I may want to push for that 100% and truly do my best even though I'm going to be spending a lot of time on tiny details. If I'm vacuuming, maybe I don't feel like moving all the furniture and something like 80% is my best. It's all relative.