Who decides what PHB variants are in play?

The rules variants presented in the DMG are clearly there for the DM to decide upon.

SOME of the variants in the PHB say that they are the DM’s prerogative. For example, VARIANT: CUSTOMIZING ABILITY SCORES (PHB p.13) says "At your Dungeon Master’s option, you can use this variant for determining your ability scores."

However, the VARIANT: ENCUMBRANCE (PHB 176) says "Here is a variant if you are looking for more detailed rules for determining how a character is hindered by the weight of equipment." AFAICT, there is no discussion of encumbrance in the DMG.

Does this mean that, RAW, individual players each decide whether or not they are using the encumbrance rule, and that a given game might have some players using the variant rule and others not?

(I realize that by Rule 0, A DM could decide that the variant rule was in play or not)