It's years since I did this crap, but the gain formula for an common emitter amp with unbypassed
emitter resistor is:- G = gmRc/1+gmRe
1 is very much smaller than gmRe , & can be ignored, so the formula looks " near as dammit" to be G = gmRc/gmRe.
Dividing top & bottom by gm gives us G = Rc/Re,so for your first circuit G=500/62, or 8.0645, just as you say.
In the standard bypassed configuration the bypass C makes Re look very, very, close to zero ohms for ac, so we get G = gmRc/1+0 , or G = gmRc.
In your second circuit, I would guess* that any additional feedback appearing in the base circuit due to the connection is minimal, because of the presence of R8, so the formula reverts to
Rc/Re, where Re is R9 & R10 in parallel, so G = 500/55, or 9.09, not 4.5.
*Only a guess-------as I said, I've haven't done this stuff for years.