While using the script $set(originaldate,%_recording_firstreleasedate%) I realized for some cases Picard don’t write the value in “ORIGINALDATE” but “ORIGINAL RELEASE DATE” and this despite showing "Original Release Date"in GUI for both cases.
You don’t tell what file formats those are or what separate software you use for inspecting the tags. But different file formats have different tagging standards. The first one is likely a file with Vorbis tags, e.g. a FLAC. There a tag with “ORIGINALDATE” is written. The second one is likely an ID3, here there is a defined frame called “TDOR”. Your other software likely displays this as “ORIGINAL RELEASE DATE”.
But Picard aims to have a single name for specific data that is then mapped to all supported file formats (if possible). So no matter how the tag is called for the original date for that file format, Picard will always use the same nice name.