Welcome!
What Mac OS is the LCIII running? I ask becasue through at least OS 7.6 and maybe OS8.0, there is a 2GB limit to any volume. A 9-GB hard drive would need five partitions. I think that limitation went away between OS8.0 and 8.1
Drive Setup, in "Utlities" on a hard drive and on most system floppies of that era, will format and partition for you.
new one SCSI 9 GB with 80
Be sure to count pins, if necessary, before getting an 80-pin adaptor. I've installed several 9G Ultra-SCSI drives in old Beige G3 servers and they were 68-pin. Yes there are 80-pin drives but are probably not as common and the 50 and 68
Internal adaptors work but there can be limitations in computers with small form factors like the LCIII (we have a Performa 475--same case). The adaptors, applied traditionally the the end of the hard drive, add length to the drive and can mke it too big for the case.
There are other ways an adaptor can be accommodated by putting it in unexpected places. Example. get an adaptor that goes on the logic board SCSI connector to allow the use of 68- or 80-pin cabling all the way to the drive. There was a site that showed how to do this but it's down at the moment.