I think storage or storage drive is the umbrella term these days. "Hard drive" was always short for "Hard Disk Drive" (which was named in comparison to Floppy Disk Drive) but since it was the only type of drive used for non-volatile internal storage for a good 20 years or so, it became a catch-all term. These days, many people understand there's two different kinds and a lot of systems have both, so hard drive is becoming recognized to mean the spinning disks; as opposed to SSD, which is now an umbrella term incorporating 2.5" SATA, M.2 SATA, and M.2 NVMe, which are all Solid State Drives but different combinations of interfaces and form factors.

Nah, the "SS" and "HD" bits refers to how each storage disk reads data. HDDs use hard metal disks to read & write data, hence it got the misnomer hard disk drive. SSDs use solid state flash memory to read & write data, hence it being called a solid state drive.

If you want the general category, you'd want to say "storage drive" specifically since if you say "drive", that can also refer to an optical drive (AKA the CD slot) or a USB drive (AKA flash/thumb drives).

I've been seeing both recently. I've opted to err on the same side and just make it clear when I'm talking about spinning rust versus solid state.