Atmospheric pressure is a result of elastic collisions between molecules, so the molecules bounce or "push" off one another. When you drink out of a straw, it's like a reverse tug of war where both sides push instead of pull. Before you start sucking, air inside and outside of the straw are at identical pressure - both sides "push" equally hard, and the liquid in the straw doesn't move. As you suck on the straw, you reduce the pressure within the straw, meaning it does not push as hard on the liquid. The forces on the liquid are now imbalanced since the air outside the straw pushes harder than air inside the straw, forcing the liquid up the straw.
Air pressure arises due to "pushes" between molecules. A vacuum does not "suck" - note there is literally nothing there to give rise to any sucking force. Rather, it is the fluid pressure that is always pushing, and if there is insufficient force pushing back, movement can occur due to the imbalanced forces.
Imagine if instead of sucking through the straw, you blow instead - the ambient air pressure around the glass doesn't suddenly change from a pushing force to a pulling force. Instead, the pushing force around the glass is simply overcome by an even larger pushing force inside the straw. But whether pressure in one area is higher or lower than another does not change the nature of the pressure force. Although pressure is fundamentally a pushing force, suction from your mouth is required to imbalance the pressure, which is why water does not get pushed up a straw spontaneously.