Youtube – Copyrighted audio disabled from YouTube video – how did they know

According to threads in the YouTube's help forums, the copyright strikes are permanent and are not removed without filing a counter-notification".

Excerpt from the page:

If you believe your content was misidentified as infringing, 
you may file a counter-notification. If you did not have all
of the rights to post the material at issue, you MUST NOT 
submit a counter-notification. 

Good luck!

YouTube reported that there was a copyright claim against my video.

A claim not a violation.

Music Publishing Rights Collecting Society is the generic name given to represent organizations that collect royalties for music.

YouTube most likely has contracts with many collecting agencies and just clumped them under one name. It's just an indication that someone out there has rights and gets royalties when you broadcast your video.

For more information on Royalties

1: http://en.wikipedia.org/wiki/Royalties#Music_royalties
2: Music royalty collection societies MCPS, PRS and PPL

And Fair Use does not work here

When a copyright holder sues a user of the work for infringment, the user may argue in defense that the use was not infringement but "fair use." Under the fair use doctrine, it is not an infringement to use the copyrighted works of another in some circumstances, such as for commentary, criticism, news reporting, or educational use. The defense generally depends on a case-by-case judgment of the facts.

On whether the Christmas song they sang is public domain (which I think only works for United States) is another thing altogether. The song could be in the public domain but recordings of the song may not be.

Yes, music copyright law is confusing and stupid.

The video will most likely not be removed but have ads slapped on it.

Consider contesting the claim with the dispute form.