Not only naming matters, but composition. For example, in USSR there were pretty rigid difference between regiment-level additional forces (artillery, EW etc) and division-level, end even army level. In WW2 some artillery detachments with specific equipment were tied directly to armies or even higher (high command), and can't be used easily from lower levels without specific reason. But it was reasonable considering communication and planning tech on these time.
Nowadays division-level tech slowly goes "lower" (from mobile artillery to recon drones), and units that can operate "separately" become smaller too. Supply also changed, it is easier now to supply less centralized groups than in past.
Anyway, army unit layout is pretty complex and interesting thing by itself.
>would not give the same cohesion and support that a Division has
As interesting fact: in first Chechen war Russian army still had "Soviet layout", Soviet officiers and overall Soviet everything (less than 5 years passed since USSR collapse). And almost-two-million army, that was designed to be near English Channel after 10 days of big war, couldn't even find any unit to do anything. So government was forced to create some diverse group from everywhere.