Yes. OSR's meaning is a little nebulous ("R" can be for "Revival" or "Renaissance" or occasionally other words), but it's more a genre of game design, tone or both. It encompasses retroclones like OSE but also (in most gamers' minds) games like Shadowdark and Troika, which are not.
Old School Essentials is a specific game, reorganizing and clarifying Basic D&D.
Necromancer/Frog God used to have a strong relationship with Labyrinth Lord (an OD&D clone) and most of their "OSR" games were presumably written for that.
The good news is that most OSR is compatible with BD&D (which is one of the reasons "OSE" sometimes gets used interchangeably for "OSR").
If you are Expeditious Retreat Press, producing 1E AD&D adventures, and I'm using Shadowdark, I can still pick up your adventure and run it with little to no conversion required, because most OSR games are extremely similar.
The only exception is when an OSR game is trying to emulate a completely different game, like Traveller, but nowadays, very few "OSR" games are doing that and the games that do tend to clearly label themselves differently. You will just have a few grognards (who are everywhere in OSR land) using the "OSR" label for non-D&D-inspired games.