It's just definition convention, really.
By the absolute, raw definition of "haptic", anything that delivers a sensation you can feel is "haptic". This includes wheel force feedback, as you say, and everything right back to the first DualShock (well... all of them) and their Eccentric Rotating Mass (title of your sextape).
DualSense isn't actually that far removed from the ERM system in DualShock, as it happens, but it uses magnetic voice coils called Linear Resonance Actuators (LRA) - similar to how mobile phones since... oooh, about 2015 have done it. They still having a moving mass, but it travels short distances along a single axis, rather than accelerating a big mass around and around, so it's quicker to respond and more detailed. In that respect, really DualSense and DualShock haptics aren't that different and it is indeed odd to say that one is haptic and the other isn't.
However, the convention is to refer to a feedback system that detects and measures the force you apply to a button or screen and delivers an appropriate tactile response as "haptic", largely thanks to the proliferation of the term by mobile phone manufacturers. Really, the term for mobile phone/tablet haptics should be "surface haptics", but nobody - up to and including specialist technology outlets - really refers to it as such.
That's essentially all that Sony means when it refers to "haptics". It's not that DualShock isn't a haptic device but, by convention of terminology, the DualSense is what most people think of as haptic.
Basically it's "thanks, Apple, for mutating the language".