OK, I've installed Simple File Manger to see whether that helps me see what you are describing. And it is different from many others, because I see that it offers me a choice of "Internal" and "SD Card" storage locations and no root view of the filesystem overall ("root" being linux talk for the base directory of the filesystem). So you probably don't have to worry about confusion arising from "/sdcard" since that's what it's calling "internal" (a better name), and what it calls "SD Card" is I'm sure the actual SD card.
(Android filesystems are complex, with many directories the user cannot access, several virtual volumes, some of which overlap giving you more than one path to the same storage location. "/sdcard" is an alias for /storage/emulated/0, which is the part of the internal storage that the user can write to. But Simple File Manager clearly tries to avoid confusion by not showing you any of that stuff - which is fine until you talk to someone like me who knows it all exists and so assumes you can see it!
).
What bothers me slightly is that it shows "SD card" on my phone too, when I don't actually have an SD card slot! However it won't let me create files there or move files to it, so it's safe, just a little disconcerting (because I'd have written it to check the filesystem existed and not show it as an option if it didn't).
The problem with "where did those files really go" is that you say the app crashed: if it crashed during the move then some part of the operation is not complete. What that might mean will depend on how exactly the app "moves" files. If you are moving files within the same storage volume that's basically a renaming operation. If you move from internal to SD or vice-versa then the app has to make a copy and delete the original. The trouble is that without looking at the source code of the app (which should be possible, it's open source) I don't know exactly how it does this, so I don't know what might happen if it crashes part way through the sequence of steps. You could code it in a way that would ensure that there's no data loss in a crash (basically have the app do what you would do manually: copy the file, verify the copy, and only then delete the original). But we don't know that they did that, and it sounds like they may not have. The worst case could leave the file deleted from internal but not written (or not completely written) to SD, in which case it's gone because file recovery from internal is "problematic" at best.
All I can suggest is restart the phone and see whether the files reappear (sometimes the phone might just get confused). If not, then try a file recovery on the SD card and see whether you can find anything, but don't hold out too much hope.
