As The Bass Player, why would I ASCAP or BMI?

I understand those agencies offer different areas of registration, such as one for the Songwriter/Composer and one for the Music Publishers. There is another category, but I think just these two apply to my question.

First off, can someone clarify those two categories for me please?

Songwriter/Composer: Does the bass player fall into this category? I have contribubuted to the songs, but just like most bass players do, by playing the bass. Some songs have actually been generated by using one of my bass lines/melodies/chord progressions and build from that.

Music Publishers: Is this equivalent to the Booking Agent / Band Promoter?

The way I understand it, if I join say BMI as a Music Publishers (to promote the band) as well as the Songwriter/Composer (being in the band), I can, in a nutshell, promote the band to say XYZ radio station so the band gets paid as a whole by the "Music Publisher" after BMI collects from the radio station, and I on the other hand get paid by the band, but that would be done through my Songwriter/Composer account at BMI? And obviously, the other band members should also be registered as Songwriter/Composer to keep it all fair and regulated, correct?

Also, I'm not sure if will actually register as the Music Publisher, cause maybe I won't want to deal with that, but would it be wise to register as a Songwriter/Composer? Do I register that band's songs (even though I just play bass in them) as this process?

The reason I ask is because I was approached by someone today asking me to lay down some bass lines for a project, and suggested that I register with BMI or ASCAP to get paid.

Now I'm confused, and just want to make sure I'm also "registered" for the stuff I've played in the band I've been with for 3 years... just in case those someday actually get some radio play.. lol

Advice?

Thank you!