This still sounds to me like a mismatch in ODBC DSN names. For example, let's say all those working reports were created by someone who used an ODBC connection called "OurData". Logicity is going to be looking for an ODBC connection on its local machine called "OurData". So long as that's there, Logicity should be able to use it with the username/password embedded in the RRD.
Now let's say you create a new report but your ODBC connection is called "MyData". The Logicity machine doesn't have a connection with that name, so it prompts. Now...we also have to consider the difference between user and system DSNs. If you have a user DSN called "MyData", it will work. But if you log into the machine under a different user account, and that user doesn't have a "MyData" connection, same problem.
I think the views are a red herring. I see a scenario where you create a report with the "MyData" connection, and it doesn't work for the reasons stated above. You then open an older report that works (possibly because it was created with the "OurData" connection?) and swap out the views. The ODBC data source is the same, so it works.
This is all very hypothetical, but there's a way you can test the theory. Open one of the reports that works and select Database > Set Datasource Location. Make a note of the data source name. Now do the same on a report that doesn't work. If it's different, you probably have your culprit.
And if it's not... well at least I didn't make it any worse.