I feel exactly as Michael and Joel do. Ludwik, if you want to show
your students that your interpretation makes sense, don't do it
with Atwood's machine. Simply show them the problem of two blocks
on a horzontal frictionless surface, connected by a string and
pulled along by a string attached to one of the blocks. That way
the same lesson comes out, and there is no confusion of directions
or conventions. The equations can be tha same as those you get
with Atwood's machine with the difference in weights replaced by
the external applied force. If it is your intention to give a
concrete example of an internal force (something which can be done
much more elegantly using Newton's third law) then why complicate
it with a pulley?