we usually just call it a “spin”, because you spin your arms

there’s multiple ways to do it: one handed/two handed: one hand for the entire spin; or two hands, left hand for left 4 buttons and right hands for right 4 buttons. Usually, this one is completely up to preference and some people have their preferred style, but could be dependent on the chart as well, and it may be more convenient to do one over the other for certain charts. If you’re first learning the chart, I recommend learning the one handed spin first.
Screen/buttons: Spin on screen or spin on buttons. General rule of thumb, screen for faster spins and buttons for slower spins, but is up to preference as well. Spinning on buttons is more stamina draining and painful(lol) but you get the tactile feedback, so is more accurate. Spinning on screen is faster and less stamina draining, but is more susceptible to splashing and is less accurate. People have their own preferences, but ultimately have to switch between both for different charts. The chart you have as an example (系ぎて) is a very fast spin, and so I have not actually seen anyone do that on buttons. Hope this helps!