The "before download" debug output shows the parameters being passed to file_downloader->get() for retrieving the JSON file, and the path and filenames are correct in both cases. Almost the first thing file_downloader->get() does is construct the HTTP GET header path using those parameters, so there isn't much opportunity for the phpBB code itself to have mangled this path after the debug output we see.
The reason the version check failed in your media embed extension case is because the phpbb.com server responded with an HTTP failure status page instead of the actual JSON version check data for the extension. Yet in the phpBB version check case you posted, your board was able to retrieve the expected JSON data.
While Paul looks for whether any clues are available for how the Media Embed extension version check failed, let's go ahead and collect information from the full "Re-Check all versions" from the ACP Extensions section, with the debug version_helper.php applied. The output will be similar to before, but there should just be a lot more of it. Cut-n-pasting that information into some
[code][/code]tags here would be great, same as before.
This additional information will be helpful, since there is at least one extension you have installed which isn't pulling its version information from phpbb.com. So it will be interesting to see if the response from that server perhaps succeeded, or holds any different clue than the 410 from phpbb.com, and/or if the responses from phpbb.com itself vary at all.