Being able to control it through an API is. What happens after you trigger the shutter is out of scope. Speaking of shutter buttons, a bunch of new cameras don’t have shutters but the button is still called a shutter button.
Also yes while yes many cameras do not have moving shutters, they still operate with an electronic shutter which just like a moving one does have various limitations.
I'd argue though to many a photographer, calling any ILC or sufficiently costly camera a DSLR is akin to a post calling an android phone a linux phone. To the layperson the nuance between ubuntu's kernel or a pixel's doesnt matter to them, however to the intended audiences it sure does.
The reflex mirror in a (D)SLR directs the image up through the viewfinder prism to the eyepiece. This mirror flips upwards just before the shutter opens. This mechanic action has a decent bit of inertia and can cause blurring in some extreme cases.
Mirrorless refers to digital cameras where the sensor operates the digital viewfinder, so like an SLR “you get what you see”. For rangefinders and TLRs, your view is offset from the picture lens, so if you’re really trying to nail a composition and not “fix it in post” SLRs and mirrorless offer an advantage, which is part of why they became so predominant.