My question is: Is it proper to send an e-mail inquiring about the
starting date or should I just sit and wait. Or does waiting show that
I am not interested enough?
There's nothing improper about either action.
Certainly you are anxious, but if it's been less than a week since your last contact, it's most likely nothing to worry about. Often, HR needs to coordinate things on their end. Sometimes that requires a lot of paperwork and discussions with the hiring manager, etc. These things can take some time.
If you want to send a quick email indicating that you haven't heard from them, and asking if there's any more information they need from you, there will most likely be no harm in that.
On the other hand, waiting at least a week wouldn't be harmful either.
Try to relax a bit. If you still haven't heard by Thursday, shoot them a note.
One time I had to wait 3 weeks (due to vacations and a lot of other hiring going on) before I got my formal offer after having informally being offered the job. It came through, we negotiated a bit more on the final details, I was hired, and it worked out fine.
One more note - don't give your notice at your current job until you get the real formal offer. It's unusual, but things could happen which derail the process.