tidal force
Why does the ocean bulge out on both the side facing the Moon and the side facing away? The answer is the tidal force, and it comes not from gravity itself but from gravity's unevenness. Because the near side of the Earth is pulled a little harder than the far side, the planet gets stretched along the line to the Moon — squeezed into a gentle football shape, with a bulge at each end.
A tidal force is the difference in gravitational pull across an extended body. Gravity weakens with distance, so the part of an object nearest the source feels a stronger tug than the part farthest away; in the body's own frame this difference appears as a stretching along the line to the source and a slight squeezing across it. Tidal forces grow steeply as you approach the source — they scale as mass divided by distance cubed — so a nearby small body can raise far stronger tides than a distant huge one. The Moon, though tiny next to the Sun, dominates Earth's tides simply because it is so much closer.
Tides are everywhere in astrophysics. They lock moons to face their planets, slowly push the Moon away from Earth while lengthening our day, heat the interiors of moons like Io and Europa through repeated flexing, distort and even tear apart stars and galaxies that pass too close, and — in the extreme — produce the 'spaghettification' that stretches anything falling into a black hole.
Jupiter's moon Io is squeezed and stretched by tidal forces as its slightly eccentric orbit carries it nearer and farther; the relentless flexing heats its interior enough to make Io the most volcanically active body in the Solar System.
Tidal flexing is a heat source powerful enough to drive worldwide volcanism.
The two tidal bulges are not 'the Moon pulling the water up on one side and centrifugal force on the other' in any simple way; both bulges arise from the same stretching by differential gravity, and the far-side bulge is real, not an artefact.