Could be possible (and if yes in which cases) that you select another "intermediate FL", even if your intention is to stop the climb at FL220?
I would figure a situation in which you did a correct read back, but you select, instead of FL220, i.e FL260 in order to get, let's say, an higher rate of climb.
I ask you this because the FL you select on board (and thereafter downlinked by Mode S transp) could be useful to understand if you are acting correctly and coherently with the ATC clearance (also when the readback have been done correctly).
In almost all cases (B744 terminology used here), the current "cleared to" altitude is set on the Mode Control Panel (MCP). I cannot think of a good reason to set a higher altitude than the cleared altitude in a climb, or lower than cleared in a descent.
Intermediate restrictions, as on a SID or STAR, are programmed into the FMS via the Control Display Unit (CDU), and are adhered to IF Vertical Navigation (VNAV) is the selected pitch mode. Those intermediate restrictions will NOT be applied if Flight Level Change (FLCH) or Vertical Speed (V/S) is used as the pitch mode. So, even if you have a readout of FMS inputs, you may not be able to tell if the automation is active to honor those restrictions.
In general, VNAV is used in climb and cruise, so all the automated restrictions and protections are applied. Occasionally V/S will be used for a cruise climb to prevent too rapid a climb and the possibility of a TCAS RA. In the descent, however, any of the 3 modes may be used at any given time, depending on the dynamics of the situation.
So, a question back to you: What info does Mode S downlink to ATC other than CURRENT altitude, position, and/or speed?