Do SSDs “turn on”?

Digital electronic equipment in computer systems all have a definite "on" state. Even if there is no mechanical moving parts in the machine, digital circuits still have various states. Every action performed by these digital circuits are essentially a transition from one state to one or more other states. If there is no power going to these circuits, they are off, which means essentially (to use an analogy) trying to get the computer to talk to it and get a response would be like trying to have a conversation with a dead corpse.

When a computer turns on, all digital circuits go through an initial state. It's typically called "reset", where all the random electronic charges in the materials are cleared out and the circuit enters a stable initial state, ready to perform its function when state changes are triggered via its input(s). This would be analogous to a HDD spinning up. It's changing from its resting state to a state of readiness.

So, yes, they have a definitive on/off state. They have to, since it's not just the storage medium but all the supporting electronics that allow us to read and write to them.

Update for the OP update

Yes, it likely uses less power when off. The thing is that it does not have to power a motor like a HDD does. A HDD will spin down when idle to conserve power (if you set it to). The only reason to keep it spinning is so that you don't have to wait for it to spin up before you can read data, which can take a second or two. But with a SSD, there aren't any concerns with things like that. It will consume power so that it may accept read and write requests, but it likely won't be consuming as much power as when it's processing read and write requests, simply because it is using less circuitry, and therefore, using less electricity. It will not be using any electricity when turned off, because (like all the other components in your computer), the motherboard would have told your power supply to cut off electrical current to the device. When turned off, unless you have a very old computer, the only electronic device in your machine to receive power will be the motherboard, and only just enough to monitor the power button. The power output to the drive will be outputting very little, if any, electrical current, in this state, though.

They definitely turn on. Specifically, the disk controller is required to be on and active.

You'll notice this when you google the infamous Sleep/Hibernate BSODs. Sandforce controllers were particularly prone to this. What would happen is sometimes the disk controller would be shut off during sleep, S3, and never wake up, forcing a reboot. In other cases, a reboot didn't even do that, forces an RMA.