"Drive" is synonymous with "transport". Both actually refer to the motor that drives the spindle of the platters. Same goes for tape drive or tape transport: a motorized mechanism for the reel of magnetic tape.
A solid state drive, SSD, does not have a motor, but is still called a drive because it is the functional equivalent of an integrated HDD and can substitute/replace a HDD. This broadening (or bastardization?) of the meaning of drive is for marketing convenience rather than maintaining semantic accuracy.
A CD/DVD drive contains only a head that writes data to the surfaces of a CD/DVD and reads it from them, not the storage device i.e. CD/DVD itself.
You forgot to mention that a CD/DVD drive also includes a motor to spin the disc! If the disc does not spin, then the R/W mechanism is useless.
A hard drive contains both the storage device i.e. a hard disk (i.e. platter), and a head that writes data to the surfaces of the disk and reads it from them.
You forgot to mention that a hard (disk) drive also includes a motor to spin the platters! If the platters do not spin, then the R/W mechanism is useless.
The all-in-one HDD that you are familiar with is a rather modern device. Some of the first disk drives I used were the size of a floor-standing dishwasher costing over US$20K. The storage modules (more commonly called disk packs), the actual recording platters, were a completely separate component from the drive, just like a reel of magnetic tape. The disk controller would be connected to the drive with bulky cables. The controller, the drive and the disk pack comprised the integrated HDD you can hold in your hand today.
As things evolved, IBM tried moving the R/W heads out of the drive into the (removable) disk cartridge for better alignment and dust sealing (IBM 3348 Winchester modules and 3340 drive) in the early 1970s. As data density increased, platter diameter was reduced. Sealed platters with the drive started to become widespread with 8" HDDs. When PCs started to use 5.25" HDDs (e.g. Seagate ST506), economies of scale put the manufacturing cost of HDDs on a steady decline. Eventually economics and performance requirements led to the integration of the controller with the sealed platter(s) and drive, the IDE hard disk drive.
But in a narrower sense, a hard drive can mean only the head not the hard disk.
No, not by any competent professional. I've never heard of such a definition or usage.
REVISION The usage you mention may be due laziness and/or because the HDD is being treated as a monolithic unit or subsystem. For example: "the drive seeks to the next track." Technically it's the R/W head assembly that actually moves, and it's the actuator or voice coil that moves that assembly, and the controller and comparators are involved to oversee the operation. But it is so much easier to say that "the drive is seeking", rather than specify the individual components involved in the operation. [If anyone knows what this "substitution of the whole instead of mentioning the parts" is called, I'd appreciate a comment.]
Side note: Back in the day when a data center could have a mix of fixed and removable platter disk drives, the removable platter disk drives were also called spindles (short for "spindle hub"). A CDC field engineer once mentioned to me that he visited a computer center that had "a hundred spindles", and that he did not want to maintain a site with that many disk drives.