One method is to use different profiles for the Terminator windows - for example, you could have this as the profile at ~/.config/terminator/config:
# This is a comment
[global_config]
focus = system
[keybindings]
full_screen = <Ctrl><Shift>F11
[profiles]
[[default]]
font = Fixed 10
background_color = "#000000" # A comment
foreground_color = "#FFFFFF" # Note that hex colour values must be quoted
scrollback_lines = '500' #More comment. Single quotes are valid too
cursor_blink = True
[[ssh]]
font = Fixed 10
background_color = "#0000BB" # A comment
foreground_color = "#FFFFFF" # Note that hex colour values must be quoted
scrollback_lines = '500' #More comment. Single quotes are valid too
cursor_blink = True
#exit_action = restart #Stops the terminal from closing after the command has been run
Then you could open the terminator window, selecting the profile with:
terminator -p ssh
You could also run the command directly with the -e option - e.g.
terminator -p ssh -e "ssh vps"
For this you may want to uncomment the exit_action = restart line, so the terminal does not exit immediately after the command is run.
For more on how to edit make/edit Terminator profiles, run man terminator_config - you can easily change the background colour by editing the background_color line - you will need to quote it.