Relation between low capacity battery for Powerbook G4 12" and logic board

Hi, I just got Powerbook G4 12". Currently, the remaining capacity of my original rechargeable battery has 321 mAh available, which is around 8 minutes left if I don't plug the AC adapter. While I'm still looking for new original battery for my PB, is it okay if I use the battery with having AC adapter plugged to my PB?, will it affect the logic board? or any significant component?.

By the way, here's the details information

AC Power:

System Sleep Timer (Minutes): 10

Disk Sleep Timer (Minutes): 10

Display Sleep Timer (Minutes): 5

Dynamic Power Step: No

Reduce Processor Speed: No

Automatic Restart On Power Loss: No

Wake On AC Change: No

Wake On Clamshell Open: Yes

Wake On LAN: Yes

Wake On Modem Ring: Yes

Display Sleep Uses Dim: Yes

Battery Power:

System Sleep Timer (Minutes): 10

Disk Sleep Timer (Minutes): 10

Display Sleep Timer (Minutes): 5

Dynamic Power Step: Yes

Reduce Processor Speed: No

Automatic Restart On Power Loss: No

Wake On AC Change: No

Wake On Clamshell Open: Yes

Wake On Modem Ring: No

Display Sleep Uses Dim: Yes

Reduce Brightness: Yes

Battery Information:

Battery Installed: Yes

First low level warning: No

Full Charge Capacity (mAh): 324

Remaining Capacity (mAh): 321

Amperage (mA): 0

Voltage (mV): 12275

Cycle Count: 280

AC Charger Information:

AC Charger (Watts): 50

Connected: Yes

Charging: No

Hardware Configuration:

Clamshell Closed: No

UPS Installed: No

Talking about original battery, do all original Powerbook G4 12" batteries have apple logo on them? and words like ".....designed by Apple in California" ?. Some people sell Powerbook G4 12" batteries on the internet and claim that their stuff are original (they don't have Apple logo on them), do you think it's reasonable for an original (brand new in box) rechargeable battery for powerbook G412" costs around $20-30 ?.

I'd very appreciate if someone can help. Thank you so much.