"i do not want to boot from it, only to save data on it,"
too bad for you - you can't do that on the Atari. You must boot in a DOS in order to then format and/or save data to a disk that is already formatted. This is the way of the Atari, live with it or go somewhere else.
"I have a bunch of DD, but they are used for C64 and IBM PC. And i've read a description, that DD compatibility in the case of 1050 means another DD. A special Atari format.
Maybe i should get some SD floppies? "
no need. 1050 can do SD and ED but not DD unless modified like with US Doubler enhancement. Atari called ED the "1050 Dual Density" but in reality it is a 128 bytes per sector MFM format which makes it have 1040 sectors. I and others steadfastly refer to this special Atari only format as ED for Enhanced Density so as to NOT confuse it with the DD or true Double Density format (256 byte MFM sectors). With a US Doubler upgrade the 1050 can read all three formats but still only one side at a time.
You must get a DOS boot disk and boot it first. Then Atari Artist can format a disk properly and/or read/write files to a disk already formatted.
I live in Hungary, Europe.
I know only one guy who has several Atari, i bought the 1010 (and the cable) from him, but he don't know the solution either.