Looks like your machine CMOS setup is password protected. CMOS NVRAM in these machines save settings as you can see in the errors, but security locks are stored in an extra small EEPROM which doesn't get erased across reboots or battery failures. The errors come from CMOS automated routines can't save the default values even if you have new CMOS battery, because the routines are security locked by these passwords stored in EEPROM. You first need unlock your machine to make new CMOS settings, or let the automated routines create defaults and save them to the CMOS NVRAM.

In some Thinkpads there was a jumper pad on the motherboard which you short to clear the startup password.
https://thinkpads.com/support/hmm/hmm_pdf/38zhmm.pdf - see page 6

However, the supervisor password is harder to clear. For that era of TPs, a forgotten supervisor password would require a full motherboard replacement if serviced by IBM - http://ps-2.kev009.com/pcpartnerinfo/ctstips/e676.htm

From what you've described, and since you're seeing the CMOS battery errors, it shouldn't be a Power On Password - that one's easy to solve by clearing the CMOS (remove CMOS battery). There's also a supervisor password which is stored securely on an EEPROM inside the laptop which you can't just clear, like hyoenmadan mentioned. I've had to do this with some older thinkpads - I think a Thinkpad X21 and an X30, it's common with the ex corporate ones or just in general, laptops were valuable back then. Your one is valuable now in a sense, there certainly aren't many Thinkpad 380Z laptops around these days.

You can fix it by making up a serial port to i2c reader, soldering it carefully onto the EEPROM, using the free software on allservice.ro to read the EEPROM, then clear the password.
It requires some relatively fine soldering skills but if you've got soldering flux and some soldering wick to clean up bridges you should be fine.

There's other methods like shorting that eeprom but that'll cause a checksum error like this guy did, maybe that can work too: http://www.os2museum.com/wp/cracking-a-thinkpad-755c/

But it doesn't take many more steps to clear it the proper way with the ibmpass utility that allservice provide. See this thread for details: https://www.allservice.ro/forum/viewtopic.php?t=47
It works similarly to a password recovery method from another website, though that one was paid for and is now mostly gone from the Internet. The information on Joe in Australia's website is very useful for figuring out the exact procedure to read out the EEPROM safely, which you can mesh up fairly easily with the free allservice method.
You can find that here: https://web.archive.org/web/20040803124543/ht … rd_recovery.htm
Notice that it has a dedicated page for the 380Z - note that that uses a 24C01 eeprom instead of the 24RF08, but the software for that is also included in the allservice 24rf08 reader software.

Because it needs a real serial port, use an older laptop with a serial port if you've got another around. I used my Thinkpad 240Z's Windows XP install to do it last time I had to 😀