Common usage among SysAdmins is that HDD refers to drives with spinning platters and not to SSDs. This search on ServerFault is neither exhaustive nor conclusive, but you can see in the questions and snippets of answers that SSD is not treated as a specific type of HDD.
WRT your addendum, I don't think it makes sense to speak of an "opposite" to solid state drives. Currently, there are only a few types of "storage drives:"
- hard drives (with spinning platters)
- optical drives
- thumb drives (basically memory sticks and memory cards)
- RAM drives (computer memory accessed as if it were a hard drive)
- solid state drives, similar in concept to thumb drives, i.e. flash memory packaged to appear to the computer as if it were a spinning hard drive
3, 4, and 5 are all "solid state" in the sense of using chips as the storage medium, but they're very different technologies, and then 1 and 2 are very different from those and quite different from each other.