Is there a real need for this tag, or would it be helpful in identifying approved posts that have third party watermarks? Granted, all of these posts would be deleted anyway since Danbooru frowns on them, so I'm not sure how this serves much purpose when the watermark tag does essentially the same thing. Attentive taggers who find such images would want to report these images in the Deletion Request forum too.
If it's agreed to be kept, then I tentatively propose the following be done:
Alias 3rd_party_watermark -> third_party_watermark
Alias third-party_watermark -> third_party_watermark
Implicate third_party_watermark -> watermark
I won't make a bulk request until an agreement is reached here.