The settings.json file actually does show which Teams client is installed. The file path is %USERPROFILE%\AppData\Roaming\Microsoft\Teams\settings.json.
If you open that file, you will notice right in the root of the JSON tree that it tells you which client is installed. Here is an example of a GCC-High client:
"version":"1.5.00.12969","environment":"Production","ring":"gcchigh-general","region":"pckgsvc-gov-c2-govtx-01","cloudEnvironment":"gcchigh"
In addition to the "ring" and "cloudEnvironment" values, there are many values in the Settings node of the JSON file that give additional clues about what is different with these packages. Just a few of them are listed below:
"startPagePathAppCache":"https://gov.teams.microsoft.us/"
"desktopUpdateHostName":"gov.teams.microsoft.us"
adal":{"authority":"https://login.microsoftonline.us/"
"feServer":{"hostname":"https://gov.teams.microsoft.us"
All of these settings and more point the client directly to the Gov, GCC, GCC-High, or DoD instances of Microsoft Teams, along with the shared components of the Teams and Skype infrastructures. Doing so has clear performance benefits (not to mention security) because any user of a gov cloud who logs into a commercial Teams client would have most, if not all of their traffic redirected after hitting commercial service endpoints.
Knowing the above information, it's relatively easy to use PowerShell to query the above JSON file and find out which client is installed per user:
Get-Content $env:UserProfile"\AppData\Roaming\Microsoft\Teams\settings.json" | ConvertFrom-Json | Select-Object Version,Ring,Environment,CloudEnvironment