It does seem many content delivery questions are already about the API, whereas content management questions could potentially not be about code. We could keep this consistent with configuration.
I think api adds some value in the same way configuration lets people follow things related to setup. If configuration appeals to those interested in infrastructure, maybe api appeals to programmers. I think it's a better tag than simply code. I agree with Chris--api can't represent just the Content Delivery API.
I could imagine a rule that "if it's not about content delivery, you can optionally add api to distinguish the question from non-api questions," would match the types of questions we have, but it seems silly and hard-to-maintain. Out of the options, I'd say leave api as part of a tag that better defines a question, similar to how we use configuration or 2011. If we only dealt with code, then it wouldn't add much and we could probably remove it completely.