Hi,
#2 is much more critical for the OPAMP. While in #1 both OPAMPS are continously in regulation, in #2 the left OPAMP is continously switched between OPEN_LOOP and CLOSED_LOOP. During OPEN_LOOP the output will saturate at either supply rail and even the input stage may saturate. Then when in CLOSED_LOOP it needs to go the regulated state. Careful OPAMP (circuit) design needs to be done. Especially on
* delay to come out of satuartion
* ringing time and amplitude .. the signal needs to settle within short time
Added:
In both cases the output of the second OPAMP may be used for a guard ring around the capacitor node to reduce leakage current effects. Thus you may reduce the capacitor size, which may give some circuit improvements.
Klaus