>You have to artificially delay and shorten your responses
Delaying responses in a professional setting can hamper your actual productivity though – and appear inattentive or disrespectful, not busy.
For one, there is a general reciprocity of responsiveness, i.e., the faster you respond to them, the faster they'll respond to you. And of course, that's relative. They won't necessarily mimic your response time, but your time will affect where their responses are in their normal range of response time.
I've witnessed this firsthand numerous times. A colleague will complain Bob takes days to respond, whereas I typically get a response the same day – and the inverse too. It's not hard data, but the pattern seems true.
The other part is that I see no real benefit in delaying a response. I happen to be in a legitimately busy phase right now, but I still try to be fairly responsive. The nuance is that my quick responses are sometimes just me saying that I'll respond in depth by the end of the day/week/whatever and to please let me know if a response is needed sooner.
I can imagine scenarios where that's not appropriate, but in general, that approach makes more sense to me. It still communicates that you're busy but doesn't artificially delay things or unintentionally disrespect people.