Twitter has been unable to protect/own the word tweet effectively in its use case. They tried to acquire the trademark in a round about way back in 2011 from Twittad [1], relating to a phrase ("let your ad meet tweets"), with a filing date all the way back to July 2008 [2]. If you dig for "tweet" in the TESS trademark database, there's nothing related to Twitter owning it however. My opinion, and from everything I've read on the subject, is that Twitter has zero trademark control over the word "tweet" unto itself.

They did acquire this trademark however ("subtweet"):

http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4803:kmz...

And there is a new trademark published for opposition from August 25, 2020, related to bird food:

http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4803:kmz...

[1] https://mashable.com/2011/10/10/twitter-tweet-trademark/

[1a] https://www.hg.org/legal-articles/update-twitter-finally-lan...

[2] http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4803:kmz...

I think Twitter is okay with people using the word "tweet" as it doesn't imply affiliation with the Twitter company, but still clearly communicates that it's related to their service.

Fun side-note: I actually had another service called "EmbedTweet" back in the day when Twitter didn't have their own embed functionality (yes, that was a long time ago). They listed my service on of their official help pages.

So yeah combined with the fact there are so many "tweet"-based names, I think you're safe using that term. Oh, and when creating an app in their Developers portal the require you to enter a name, and they don't flag "tweet" but I think they do flag "Twitter".