The Liechtenauer glosses are quite clear that right handers should predominantly open with cuts from the right side, left handers from the left, as these are their dominant sides.
There are two distinct approaches to cutting in the 15th century text, cutting through and cutting to the point.
Cutting through refers to taking the blade all of the way through the target and to the other side. An Oberhau for example might cut all the way through a target and fall down into an Alber type of position.
The mechanics for cutting through typically engage the larger muscle groups as a part of the cutting process, and rely less on a "push-pull" of the grip, and more on a swinging of the whole body as a single motion segment.
We see this kind of cut in the section on over-running (Oberlauffen), in which the opponent cuts too far, allowing the opponent to chase after their cut to the upper opening.
For this very reason the 15th Century writers admonish us not to make too much use of these wider cutting movements, instead cutting to the point.
Cutting to the Point
Cutting to the point refers to ending our cuts with our point very much toward the opponent, rather than cutting all of the way through the target and (for example) to the ground.
The mechanics for cutting to the point rely on more of a push-pull action of the hands which accelerates the blade quickly in a tight circle. Because we need to end on-point our cuts rarely travel far past the impact point. In the case where the opponent voids our attack this means we must also rapidly decelerate our cuts (though if they are hit or parry then their body or blade does that work for us).
Cutting to the point is seen as a superior method in the early texts as we end with our blade in a position to thrust or to set aside the opponent's attacks, though the cuts rarely land with as much force as a full cut through.