Okay, I think I have a solution. I'll post it here and also on the Synology forums. Thanks, Roger!

EDIT:
I have posted a BETTER SIMPLER solution here:
http://forum.synology.com/enu/viewtopic.php?f=222&t=85425&p=323055#p323055

Just tell Video Station to index everything as "Home Movies". Then it won't search the internet for the metadata. More details at the link I just posted.

Old information below. Do not use:

Steps to wipe incorrect metadata from Synology Video Station and start over from scratch, while also disabling the metadata search.

Problem:

- You are running Synology DiskStation 5.x on a Synology DS214play. (Note: I have tested this procedure on DSM 5.0-4482 and Video Station 1.3-0544 only. This might work on other devices or versions, but I don't know.)
- You are using Synology Video Station to manage videos.
- Synology Video Station automatically searches the internet for Metadata for your video files.
- Synology Video Station makes bad choices about the Metadata, such as (among other things) tagging innocent kids/educational films with metadata and box art for Porn movies. Woopsie daisy.
- Synology has not yet issued a fix for this problem (check for that first, before trying this procedure).
- Synology has not yet offered a menu option to turn off the automatic internet metadata searching feature (check for that first, before trying this procedure).
- You would personally prefer to just have the file names of the video files, and not bother to attempt metadata searches, and you're OK with not having any metadata at all.

DISCLAIMER:

This is the first time I've ever owned a Synology NAS, I've only had it for two days, so I'm no expert. I don't actually know if this will work correctly for everyone. I know it worked for me, that's all. My NAS is set up bare-bones so far, so if you have other software on your NAS, I don't know how this will work for you. Although I have Plex and it's still working OK. Mainly, the worry here is that this involves modifying files on your NAS that Synology doesn't expect you to be modifying. Do not do this procedure unless you understand each of the steps and are willing to take the risk of possibly rendering your NAS nonfunctional or losing important data on the NAS and having to reformat the whole thing.

Steps to fix the problem:

1. Have a tool on hand which allows you to edit the files on the hard disk of the DiskStation remotely across the network. In my case I used the freely available and very popular WINSCP tool to do this. You can use whatever tool you're comfortable with. Note that you will need to configure the DSM to allow this, the setting is under Control Panel, Terminal & SNMP, "Enable SSH Service".

2. Write down your existing settings for the Video Station program. Go into Video Station, hit Settings, and write everything down, including the links to the folders on your disk which are currently being indexed for videos. Write down the permissions you assigned in the Privileges tab. Write down any other settings you care about.

3. Go to Package Center, and deinstall Video Station.

4. Important: When deinstalling Video Station, make sure to tick the tickybox that says to "Remove Video Station Database". The reason you are deinstalling Video Station is to stop Video Station from running, to clear out all of its folder links which collect and index video files on your hard disk, clear out its permissions list, and to allow it to cleanly get rid of its existing database (full of bad metadata) so that we can start over from scratch. This is the simplest way I know how to do this. If you have ways of easily doing those things that don't involve deinstalling, feel free to do that instead.

5. Log into the DSM using your tool (WINSCP or whatever you have). When logging in, use the username of "root" and use your admin password. Using your tool, remove the folder /volume1/@appstore/VideoStation if it is not fully removed already. (Again, if you know how to make Video Station forget EVERYTHING without doing this step, feel free to do that. I had to do all of these steps or else it kept re-adding the existing metadata from the old Video Station database.)

6. RESTART the DSM now to make sure everything about the Video Station is cleared out of memory.

7. At this point, if you had placed video files in the default "video" folder on the DSM's hard disk, you might possibly need to move them temporarily out of that folder to prevent the Video Station reinstall from automatically re-indexing everything. I didn't have to, but, I deliberately had not put any videos in there to begin with. All my videos are in separate user folders which I'd had to link separately. Feel free to try these steps WITHOUT moving the video files first. It might work without moving the files, not sure.

8. In Package Center, reinstall Video Station. DO NOT RUN IT YET.

9. Hopefully it will not automatically begin indexing your video files if you haven't run Video Station yet. I'm not certain if that's how it works, but I think so. That's the way it worked on my system.

10. Using your tool, navigate to /volume1/@appstore/VideoStation/plugins folder.

11. You will need to edit two files. In the subfolders "syno_themoviedb" and "syno_thetvdb", there is, in each folder, a file called "search.php". That is the file you will edit in each folder.

12. At the top of each of the two "search.php" files are some variables and URLs which define where it gets its search results. Find those.

13. For the variables "API_URL" and "BANNER_URL", mangle the main name of the site, the part right before ".com". For example, if it says "http://www.thetvdb.com/api/", change it to "http://www.NOPEthetvdbNOPENOPE.com/api/" or something like that. Anything to make the URL fail utterly.

14. Save both of the files you edited.

15. Using your tool, navigate to /volume1/@appstore/VideoStation/sbin

16. Locate the file synovideometadatad which is the metadata gathering daemon.

17. Modify the file properties and turn off all three EXECUTE PERMISSONS on that file. If you are using WINSCP, here is how that is done: Right click on the file and select PROPERTIES. There will be a PERMISSONS section with OWNER, GROUP, and OTHERS for that file. Each of those rows will have a tickbox for Read, Write, and Execute permissions labeled R, W, and X. You are going to untick the "X" tickbox for all three. Save those changes by pressing OK.

18. The "rights" on the file synovideometadatad should now read "rw-r--r--".

19. RESTART the DSM again to make sure it is no longer running the synovideometadatad daemon that you've tagged as non-executable.

20. If you moved files out of the main "video" folder at the start of this process, you can move them back now.

21. Run Video Station from the DSM's main menu.

22. The first thing it will ask you to do is to add your folders that contain your videos. Go ahead and do that, using the stuff you wrote down at the start of this procedure. Also go into the Privileges tab and set your desired privileges for your users. Go through the rest of the settings panel and put all your settings back the way they were.

23. In theory it should re-index and all should be well, no more bad metadata, just file names.

24. Go into the Package Center and under Settings, Auto Updates, set it so that it does not auto-update Video Station. (If an update becomes available, you can update it by hand if you are willing to make the above modifications all over again when it updates.)