Twitter does not allow monitoring of its API (opens in new tab)

Not sure I understand the issue. If you're regularly using the API then you're monitoring it. You don't need to make extra requests to also monitor it. You should build your application to alert you if requests to the API start failing, maybe even build in a graceful retry system and a message to your customers that you're experiencing issues outside of your control and your services will be back shortly.

Add a queue (you already have one right?) so that any messages sent through your api will still get out and you've provided as good of customer service as you're going to get when relying on a 3rd-party API.

I imagine Twitter also wouldn't want all of the sites using it's API to have a message saying "Twitter API is Down", that would be unfortunate for them. I think that also explains the wording of that term.

Sounds suspiciously like they want to avoid having people slam their APIs unnecessarily. They probably don't care how you use their APIs, they just don't want you constantly polling them and/or sending of query bursts just to get some latency stats. Every API hit adds to the load on their servers, and it may also mess up their own internal API statistic gathering.

So they probably want some pre-established legalese that allows them to kick people off who make an unreasonable number of queries for no practical reason.

One developer doing this probably isn't a concern to them, but if thousands of people all over the place do it, that could become annoying.