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.