"The tone was obvious"

Written slips are the worst, because they arrive with no face and no voice and you supply both. "Please send the figures by Friday." Read it in a flat voice and it's an instruction. Read it in a cold voice and it's a rebuke. Read it in a tired voice at the end of a day of forty emails and it's someone typing as fast as they can. The words are the same, and the tone you hear is the tone you brought.

Justin Kruger, Nicholas Epley and their colleagues ran five experiments in 2005 on how well people communicate tone by email. Senders were confident their sarcasm, humour or seriousness would come across, and it came across much less often than they thought. The reason, in the last two studies, was that senders couldn't detach from their own perspective. They heard their own message in the tone they intended, and assumed the reader would hear it the same way. Someone typing "Please send the figures by Friday" hears it as brisk and ordinary. They have no idea it's about to be read as cold.

This cuts in both directions and you need both. First, the sender of the blunt email almost certainly didn't hear it as blunt. They heard their own intention, which was probably "get this off the list before the next meeting". The coldness was added at your end, and the sender doesn't know it exists. Second, you have sent that email. You've sent dozens, and you'd be startled to learn how some of them landed, because you heard them in your own voice too.

I'd put it this way, and it's my interpretation. Tone in writing isn't sent. It's assigned by the reader, and the reader who's already made a charge assigns the tone that fits it. Once you've decided a manager is dismissive, every short email is evidence, and there's no email short enough to be neutral. The words haven't changed. Your assignment has. The practical rule for written slips is the plainest one in the book, and it's this. Read it once in the most boring voice you can manage. If it still stings in a boring voice, it might mean something. If it doesn't, it didn't.