TMSLogger is declared in units VCL.TMSLogger or FMX.TMSLogger, you should use one of those units - or simply TMSLogger if your application is configured to use VCL or FMX alias.
If you use only TMSLoggingCore, then you should use TMSDefaultLogger global variable.
I'm not sure what exactly are you referring to regarding Linux in documentation?
As it also works when targeting Windows, the way to be able to share code between platforms would be to use TMSLoggingCore and TMSDefaultLogger everywhere instead of VCL.Logging and FMX.Logging and TMSLogger ?