MusicBrainz is open source, and most of the committed users here aren't big fans of copyright!
Freso already posted this in another thread:
That said, I think you don't have to worry too much about the copyright anyway, it should automatically apply.
Here's a NZ site with some useful information:
And a US one that says pretty much the same thing:
As far as I can tell 'Songrite' simply stores your songs/lyrics so that you have proof that you've written them at a specific date. Uploading your song and lyrics to Bandcamp does exactly the same thing - that website has a time stamp of when you uploaded the music too.
If you want to go really official, this seems to be the way to do it in the US:
Registering a copyright with the USCO costs $35, as of 2011. However, for this fee you can register any number of lyrics, as long as they were all written by the same person.
Which probably seems like your best, 'most official' way. Probably not that complicated either:
ASCAP or BMI do kind of similar things, registering your music, but they also control royalties/radio play etc for you.
I've just googled this now, but this seems like an interesting post:
Enjoy going down that endless rabbit hole!
Anyway, personally I would either relax and just leave it, or do it in the most direct way possible (go directly to the USCO, or your local equivalent - NZ has some very helpful govt offices that I would call over here, you might do so too).