ALS and ADS, what does it means?
Advantage essentially has two modes of operation: LOCAL and REMOTE. The AIS (Internet) option is really just another flavor of REMOTE; you can probably ignore that option until you get other things figured out.
The local option (ADS_LOCAL_SERVER) does not necessarily have any ties to the local system (localhost). It is a mode of operation where the client application opens the tables directly (much the same as if you open a file in an editor). For this to work, the client application has to be able to get access to the file through the file system (e.g., on a local drive or on a mapped drive such as through the net use command). So you could use ADS_LOCAL_SERVER to open a table on another machine if you can map a drive to that other machine.
The remote option (ADS_REMOTE_SERVER) is the true client/server option. This requires that Advantage Database Server be running on the machine where the tables are located (this is a simplification but for your situation it is probably correct). With ADS_REMOTE_SERVER, the client application does not need direct file access through the file system to open tables. It simply needs to be able to communicate with Advantage Database server through the TCP/IP stack (sockets).
If you want to use the remote option with ODBC, make sure the setup dialog for the DSN has only the "Remote Server (ADS)" option selected. And then make sure that the path for the database is correct.
For a remote connection, the best solution is probably to use a UNC path to the location:
Or possibly consider using the IP address and port to avoid possible problems with discovery: