It's much better to do what your lawyer said, in this case. Strictly speaking, the two setences do not equal in interpretation, and may easily cause confusion at CIS side who may then ask you to supplement additional proof anyway and generate delay and inconvenience.
It shouldn't be difficult for your to obtain such a letter, as far as it states the truth. You may also seek help from the office of international students or your professor.
Alternatively , you may submit application now with what you have, and send in the additional document later ASAP along with the receipt notice copy, before USCIS asks. Or, simply wait until Aug if you can until you get everything you need.