While they may have spent time with different basses James Jamerson and Carol Kaye are both most associated with Precision basses. You may find a particular string type (ie flatwound) will give a more retro sound.
I most associate Macca with the Hofner violin bass , couldn't really advise on how to get that sound.
The main issue for me is the scale length, I don't like short scale basses, I'd guess that I'm in the majority, that's not to say that it isn't for you.
My gut instinct is that the Precision is the more versatile bass, shop around , there are many variations, you may find a lightweigh precision.... or try an SG and fall in love ... or ....