Drive name? What is the correct term for the "sda" part of "/dev/sda"?

sda is the device name. /dev/sda is the device path.

Think of /sbin/fdisk, fdisk is the file name, while /sbin/fdisk is the file path.

The sda part of /dev/sda may represent several things depending on your point of view:

1. File name
   In general, it is the name of a file, which, if it is a disk, is a block device, therefore, a device name

   As root:

   $ ls -la /dev/sd*
   brw-rw---- 1 root disk 8, 0 Sep 11 22:01 /dev/sda
   brw-rw---- 1 root disk 8, 1 Sep 11 22:02 /dev/sda1
   brw-rw---- 1 root disk 8, 2 Sep 11 22:01 /dev/sda2
   

2. Disk names
   sda is the string that gives a name to a disk (diferent than partitions sda1, sda2, sdX, etc. ). It is usually generated by udev based on the applied device rules.

3. Directory path
   sda is the basename of the path /dev/sda

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Limited to the interpretation related to disk names:

In the old times: sda used to be the device name of SCSI disk a.

As the same library got extended to SATA drives I guess that now it should be called:

device name: SATA/SCSI/SAS disk a

Or simply:

device name: first SATA/SCSI/SAS disk

It is kernel-generated block disk/partition device name:

sda is a disk name generated by kernel. Kernel drivers (including SCSI stack which is happened to be very convenient to use for SATA disks) fill disk_name field of gendisk structure (i.e. for SCSI: drivers/scsi/sd.c#L3338) to generate sda name.

This name is later used to be a name of corresponding block device in /dev, /sys and /proc/{partitions,diskstats}. However, manual for procfs names it partition name (proc(5)) and documentation on disk stats calls it device name (iostats.txt).