I think this bass sound comes more from the bass it self, cab/di and preamp.

Are you playing the instrument or are trying to do it with a VSTi ? Trillian is one of the best for bass...

I'm trying to recreate it with a VSTi (Logic). I don't have any instruments. I was under the impression that I should be able to edit the parameters of a pre-defined bass instrument in Logic to get to the sound I'm after.

I don't use Logic. I'm sure someone will step in with more detailed.
Most vsti start from samples, recordings of a real instruments. I hope you can reach the sound you want but it may not be possible with just the tweaking of parameters.