This has been an ongoing problem with the new launcher (2.0.x, 2.1.x).  There is no method provided to obtain the launcher betas.  Every time there is a launcher beta, the launcher notifies that there is an update, but the link provided goes to the regular download page for release versions.  When the launcher is downloaded, it is the exact same release version that is already installed.

Steps to reproduce:

Open launcher.

See "Launcher update" notification.

Click "click here" link.

See https://www.minecraft.net/en-us/download/ open.

Click ".deb package".

Open Minecraft.deb with Engrampa to check version in /DEBIAN/control .

See that version is the same as current installation.

Open Minecraft.deb with gdebi anyway just in case.

See "Same version is already installed".

Click "Reinstall Package" just in case, because why not.

Open launcher.

See "Launcher update" notification.

Check Settings -> About.

See that launcher version has not changed.