Three separate things people call "format"
Container. MP4, MOV, WebM, MKV. How the streams are packaged. It carries no colour information of its own.
Codec. H.264, HEVC, VP9, AV1. How the pictures are compressed. HDR10 delivery normally uses HEVC Main 10, because that profile supports 10-bit.
Colour signalling. Primaries, transfer, matrix, bit depth. This is the part that makes a file HDR or not.
An MP4 is HDR when the stream inside it carries BT.2020 primaries and the PQ transfer at 10-bit. An MP4 with bt709 at 8-bit is SDR. The extension is identical in both cases.
Just read the stream
ffprobe -v error -select_streams v:0 \
-show_entries stream=codec_name,pix_fmt,color_primaries,color_transfer,color_space \
-of default=nw=1 input.mp4
An HDR10 file reports hevc, yuv420p10le, bt2020, smpte2084, bt2020nc. Anything reporting bt709 in the transfer field is SDR whatever else it says.
Renaming does nothing
Changing a file's extension changes nothing about its contents, and neither does remuxing MP4 to MOV or the reverse. Both operations move streams between containers without touching the colour signalling. If a tool claims to make a file HDR by changing the container, it is either mislabelling an SDR file or doing something else it has not told you about.
Mislabelling is worse than doing nothing: a file tagged PQ that contains SDR content will be interpreted by the display against an absolute-luminance curve it does not match, and the result looks wrong in the specific washed-out way described under tone mapping.
What is accepted here
MP4, MOV and WebM, up to 60 seconds and 500 MB. Output is always an HDR10 MP4 at the resolution and frame rate you uploaded. See what the converter outputs.
Questions people ask
Does renaming the file to .mov change anything?
No. The extension and the container carry no colour information, and remuxing between MP4 and MOV moves streams without touching the colour signalling. If a tool claims to make a file HDR by changing the container, it is either mislabelling it or doing something it has not told you about.
Is MOV better than MP4 for HDR?
Neither is better. Both carry HEVC with HDR10 signalling perfectly well. MOV appears more often in professional workflows and MP4 in delivery, which is convention rather than capability. Output here is always MP4.