Is it best practice to install client tools (like Oracle SQL Developer) on the database server or another machine

Each job step has a setting that allows it's output to be sent to a text file. This can be cumbersome because it's a text file located in the file system and not in a table and might be hard to get to. There is also an option to send the output of a job step to a table. You can query the table, but it's still clunky. (It is hard to write a query to pick the "X rows affected" string out of a blob of text, then pull the number out so that you can aggregate it, etc.)

I would recommend that you build a simple table in your user database (not in msdb or master) to keep that information. IMO, this is the best way to keep close track of the number of rows affected by each command, particularly if you want to be able to write simple queries to do historical reporting. This way, you can decide to log at the job step level, or if the step calls a stored procedure with multiple commands, you can log a audit record after each command. As always, @@ROWCOUNT is your friend.

SQL Developer is using the so called "thin driver".

For this the JDBC driver (a file named ojdbc6.jar) implements the complete Oracle network protocol which is usually handled by the Oracle client. It could use the another implementation: the "OCI" (or "thick") driver which would require an Oracle client to be installed. That would be necessary if e.g. Transparent Application Failover in a RAC is required.

This is not something specific to SQL Developer but to any Java application using JDBC to connect to the database.