Rithmic asks for two things beyond your login: which system your account lives on, and which gateway to reach it through. The system is the same name you pick on R|Trader Pro's login screen. Tradeus shows Rithmic's own list of systems: click the field and pick yours, or type the first letters to find it.
If the firm you picked under Prop firm issues Rithmic logins, the system box opens already searching for it, by a word from the firm's name such as Apex; a name you typed under Other firm… is searched as you typed it. A firm whose list has no Rithmic, reached through Show all brokers, leaves the box empty. Either way the search narrows the list; it does not choose for you. Open it and pick your system, because the exact name is Rithmic's, and Connect Rithmic account stays greyed out until a system is picked. The connection is then filed under that firm — see Connect a prop firm account.
The Gateway list is 13 regional entry points — Chicago, Frankfurt, New York, Sydney and so on. Not every gateway carries every system: Core (Chicago) lists the most, and the system list follows the gateway you pick. Core (Chicago) is the default; change it only if you have been told to.