Etiquette question: is there such a thing as accepting an offer 'too early'? TLDR: Got a PhD offer at top choice, all details checked out, and I want to accept now (much earlier than the April 15th deadline).

A colleague (at the professor level) I mentioned this to said I should wait as it could look 'too earnest' accepting roughly 10 days after notification. 

Is there any truth to this claim? I.e. does it 'look bad' (i.e. desperate) if I accept now? I'm just genuinely excited and am ready to lock it in.

Additional details (from comments):


This offer is a 'hard' offer such that it is from the department, many cc'd people, and is in writing
I have other 'hard' offers from other universities and have yet to hear back from a few more


Other relevant questions:


Responding to a prof&#39;s email about acceptance in a PhD program


Update: 

Thank you all SO much for your input. I went ahead and accepted the offer and got feedback saying they appreciated the rapid response and the PI was eager to arrange a talk for next steps. While this is likely the standard positive response, I have marked Buffy's answer as correct. The first two sentences made the most sense: 'Actually, it is simple courtesy to accept as soon as you know you will do so. This helps the institution deal with its obligations to you and others.' I then let the other universities know immediately after and I hope this helps their search process.
Actually, it is simple courtesy to accept as soon as you know you will do so. This helps the institution deal with its obligations to you and others. It isn't a game you play for "advantage". You hope to develop a long term relationship with this institution. Treat them with respect and expect them to reciprocate. 

Don't accept before you are ready, such as when other offers might come in, but don't delay for any perceived "strategic" reason.  

Similarly, once you have accepted the offer, extend the other universities the courtesy of letting them know you will not be accepting theirs.