The three move sequence that you were taught sounds like it is a basic issue/open/close drill. I have had teachers tell me that these moves are the most important for staff/spear, so it would be a good place to start learning staff/spear prior to or along with learning forms. I suspect that all styles probably use these three basic moves. I learned them for Yang style, Chen style (where they are practiced both individually and sequentially as “pole shaking” power training), and Fu style.
Note that while the form that Vincent Chu demonstrates uses a short staff, the grip is more spear like, i.e. held with one of the hands at the butt end of the shaft (as I understand it, and as I have been taught, staff would typically be held in thirds, with both forward and back ends extending about equally - although the attacking end is often allowed to extend farther than the rear portion, especially in thrusts, where the shaft often slides through the forward hand). I would consider Chu to be demonstrating a spear form using a staff. I understand that it is very common to use spear techniques and forms for staff forms.