[Hendrik Magar]

Problem is in pixel dimensions.

full HD is 1920×1080 (pixel aspect 1)
1080/60i is 1440×1080 (pixel aspect 0.75)

It is the same thing as in PAL and NTSC.
For example. Editing in PAL you always have the same framesize 720×576. But depending on a show (is it 4:3 or 16:9) we change pixel aspect, because the framesize is fixed. If you make 720×570 square pixel (pixel aspect 1) you wount get a 4:3 image nor 16:9. For 4:3 pixel aspect 0.9375 is needed and for 16:9 we need 0.7031. It means that pixel isn’t a square, it’s a rectangle.

NOW BACK TO YOUR CASE.
Full HD has square pixels. You have rectangle. FCP detects that and shows the right 16:9 image. Most of computer based playback systems do not understand it and they always play on SQUARE PIXEL MODE.

So make sure what kind of pixel dimension is required for your show. Then put the right settings while exporting (use compressor for exporting – there you can easily change all needed settings). Later on you can check the file inside FCP if the dimensions are right. Just import the exported file into FCP.

If you have to show a preview file to someone through computer, then just make a square pixel proxy file with compressor. (in inspector, under geometry tab put pixel aspect to square and type in some regular 16:9 framesize…..1280×720 or 640×360 or smt else)