Not a habitat as such, but the first inflatable human-rated facility if you like was on Voskhod 2, equipped with an inflatable airlock.
Otherwise, as able to sustain human life as a berthed module of the ISS, BEAM is indeed the first inflatable habitat.
Everything else is as you say.
The BEAM module was developed by Bigelow Aerospace and installed on the ISS as the first
ever human occupied inflatable module.
First Inflatable Habitat for Astronauts All Pumped On Space Station
1st Inflatable Habitat for Astronauts All Pumped Up on Space Station
An astronaut just deployed the Space Station's first inflatable habitat
Astronauts enter world's first inflatable space habitat (Update)
First Ever Inflatable Habitat Attached To The International Space Station
sub-scale inflatable space module test: