How is it possible to start processes that are actually visible on the
Windows machine?
A good start would be by creating the process in the session of the currently logged on user. That's usually session 1 if there's only one person logged on. If multiple people are logged on, it might be session 2 or 3 or 27. You'd have to run code in wtsapi32.dll first to find who was connected to which session. In modern versions of Windows, session 0 is reserved for services and system functions.
You're starting a process in session 0, therefore no one can see it.
psexec can start a remote process and let you choose a different session in which to start it, but I don't know of any Linux equivalent. I looked at the man page for winexe and it does not appear to have that option.
Edit: Microsoft's official stance is that launching interactive processes remotely is too big of a security risk, and so they inhibit your ability to do it... but we can still work around it if we're willing to get dirty:
schtasks.exe /create /S COMPUTERNAME /RU "NT AUTHORITY\SYSTEM" /RL HIGHEST /SC ONSTART /TN "RemoteProcess" /TR "program.exe \"argument 1\" \"argument 2\""
schtasks.exe /Run /S COMPUTERNAME /I /TN "RemoteProcess"
schtasks.exe /Delete /S COMPUTERNAME /TN "RemoteProcess"