Your partitioning between those two things is good, but I still don't think that either label applies to them:

> Steam does control the vast share of desktop gaming.

Between the Epic Games Store, GOG, Humble Bundle, Xbox, Origin, Itch, and a few others, I don't believe their control is anywhere close to the fraction needed for Steam to be a "monopoly", either legally or in casual speak.

> Steam does control

...and, what's more, they don't "control" anything - what prevents you from either using multiple clients (on the player side) or selling on multiple storefronts (on the developer side)?

A monopoly has to monopolize some limited resource or market - you can't really have a monopoly if there's no limiting or exclusivity. That's like saying that Fortnite is "monopolizing" the battle royale genre because it's the most popular - it is the most popular, but there's no exclusivity because you can always play another battle royale in addition to Fortnite.

Monopolies need pie charts (limited resources that are taken by a single actor), but Steam is a bar in a bar chart.