After digging a little bit, I found the source of the problem, which is a peculiar way of Global Mapper's transferring projection from the dialogue box to the actual file.
After numerous attempts (exporting different area sizes) I managed to make it work. I noticed that this time the GeoTIFF had already contained data on elevation and the area size - these things were previously set to default values (0-100 etc). I checked the prj and twf files, and it was immediately obvious the solution must be there, as the size of prj file was almost double than in the case of the unsuccessful attempt. Upon listing the file, everything became clear.
The old (failed) prj file looked like this:
/beginning of file/
GEOGCS["GCS_WGS_1984",DATUM["D_WGS84",SPHEROID["WGS84",6378137,298.257223560493]],PRIMEM["Greenwich",0],UNIT["Degree",0.017453292519943295]]
/end of file/
Whereas the new one contained much more information:
/beginning of file/
PROJCS["UTM_Zone_51_Northern_Hemisphere",GEOGCS["GCS_WGS_1984",DATUM["D_WGS84",SPHEROID["WGS84",6378137,298.257223560493]],PRIMEM["Greenwich",0],UNIT["Degree",0.017453292519943295]],PROJECTION["Transverse_Mercator"],PARAMETER["latitude_of_origin",0],PARAMETER["central_meridian",123],PARAMETER["scale_factor",0.9996],PARAMETER["false_easting",500000],PARAMETER["false_northing",0],UNIT["Meter",1]]
/end of file/
The new twf file also contained different data from the old one.
The old file:
/beginning of file/
0.00029296875000005
0
0
-0.00029296875
120.40219999999999
23.3922
/end of file/
The new file:
/beginning of file/
32.613132068
0
0
-32.613132068
234468.58048
2589328.152
/end of file/
So far so good, but I still had no idea what happened, as I didn't do anything different than before. The only new thing I did was that I exported a bmp from MapQuest OSM server via Global Mapper (thinking that somehow things may align themselves, coming through the same machinery). Coincidentally, the BMP raster/image export also happens to offer prj and twf files, so I checked them (why not?). These files were the ones with the correct data.
However, to my frustration, I wasn't able to replicate the results, and I found an only slightly related video on youtube that suggested manual setting of projection in Global Mapper's setting. And there was the solution of the problem. The default projection setting in Global Mapper is 'Geographic (Latitude/Longitude)", which must be changed to UTM (the zone will adjust itself automatically). After this, all attempts of exporting different areas were successful.
I should have probably spent more time experimenting with exporting, yet the solution came by pure accident (how could I predict Global Mapper's unpredictable behavior while setting the projection?), and I might as well have missed it. Anyway, I hope it may save others a little time.