I have heard of chemical/strangling/parasitical type competition. The banyan tree is territorial, for instance.
But we would need another name, other than territorial, carnivorous or vegetarian, to describe plant predators which overtly, actively fed on the physical structure or leaves of fellow plants.
It is likely that there are much more parasitic plants than carnivorous plants.
Plants that feed on other plants must do it similarly to a fungus, by penetrating them and growing into them a root-like organ, for sucking their fluids.
A plant could not bite and chew another plant, because, like the fungal cells, the plant cells have abandoned their ancestral animal-like mobility, by covering their cells with walls made of cellulose, which prevent cell mobility. While there are a few plants capable of infrequent fast movements, like the Venus flytrap, they use special tricks for creating tension in an elastic structure, like when drawing a bow, which would not be suitable for sustaining a sequence of movements.
I would think capabilities like that would be recoverable, if the biological economics worked.
But your point that parasitical plants continuously live off other plants, i.e. they essentially farm them, resolves that. Given victim plants can't run away, their metabolisms are worth far more than any one-time resource extraction.
I guess to effectively live a long life by eating other stuff, you need to be able to move, or what you eat need to be able to move to you.
I suppose you could view the passive offspring dispersal system (wind, current, animal digestive tract, raindrops, etc.) as a form of intergenerational movement.