I have 5 actions, 2 of them are named as
"Run with item". But It feels like I have 4 actions, but one of them has two options -
"Run with item". Depending on which action will be selected as active in the Action Editor, which will be exported along with the Armature. Both animations cannot exist in the same file.
There are interesting patterns that might indicate a problem. I have spotted and shown them in this short video.