I verbally counter offered by phone, i have not heard back so i assume they are either collecting their savings and trying to convince their parent guarantor to submit to rentors insurance and a background check...
I would always inform them that it's first come, first serve. That does 2 things: 1) gets them moving quickly if they want to rent the property, and 2) gives you an easy out (not that you need it legally) to rent to another.
YET, i would like to know if it is my responsibility to do this in writing at this point in case they have some reason to accuse me of not clarifying the need for the guarantor to submit to a screening?
You should inform them that they have to go through the application process and get approved by you. That doesn't have to be in writing - a verbal acknowledgement of such is acceptable.
You can just email or text message them if you feel uncomfortable by not giving them as early a notice as possible.
But that 1st tenant doesn't seem to be operating to quickly or really wanting the apartment it would seem OR they would have immediately supplied the information as requested OR they have issues (then you don't want them anyway).