I'd say "deadband" is the more end-user term, and "hysteresis" more of a engineering term. Engineers will understand deadband in this context, but Joe Sixpack or even a electrician installing a thermostat may not understand what "hysteresis" means.
I disagree that these words apply to more complex processes. What is going on in a simple thermostat is exactly hysteresis, which is fine to express as deadband to laymen.
You are overthinking this.
The more precise meaning of deadband is actually different from hysteresis. A deadband is a region where the system doesn't respond to changes. This does occur in hysteresis, but the general case does not require hysteresis.
For example, suppose you have a proportional temperature controller instead of on/off like what the thermostat is doing. Let's say it has a 1° deadband and the temperature is slowly rising from 65° to 70°. This will cause a decreasing control response. If the temperature now drops, the control response won't change until it gets to 69°, then will increase with decreasing temperature again.
Such a deadband is usually undesirable, so systems are specified not to exceed a particular deadband. This is common in mechanical systems due to the combination of looseness of fit and static friction. It is often called the "backlash" in gear systems. Electronic systems can have a deadband caused by two diode drops, for example.
So in summary, it is OK to call the difference between the two hysteresis thresholds a deadband, but not all deadbands are due to hysteresis. In common usage, this is rather a fine distinction and I wouldn't count on people realizing this. Usually what is meant by the deadband in a system is clear enough from context.
In a engineering setting, use "hysteresis" when it really is hysteresis. In a lay setting, use "deadband" and don't expect anyone to understand nuances of hysteresis versus some other cause of a deadband.