@hornetrider Please post your full config.g and RepRapFirmware version (send M115).
My guess right now is that the x-axis driver on the board or the wiring to the motor might have an issue.
To test, you could swap X and Y motor wiring and see if the problem swaps to the Y axis. You will need to update the driver assignments in config.g: assuming your M584 is something like M584 X0 Y1 Z2 E3, swap the assigned driver for X and Y, ie M584 X1 Y0 Z2 E3.
Is there a way to run a diagnostic procedure to find any issues?
For diagnostics, run a print until you get some loss of steps, then send M122, and post the response.
In case the driver should be the problem, is there a possibility to use the E1 driver instead for the X-Axis?
If the E1 driver is free, then yes. Again assuming your M584 is something like M584 X0 Y1 Z2 E3, change the assigned driver for X, ie M584 X4 Y0 Z2 E3 and move the X motor wiring to the E1 driver connection.
Ian