Good advice in this thread.
I can't speak to your specific problems or what you're looking to hear, but I've found that for me, "clean bass" falls into two categories: muting unwanted noise (this isn't just ringing strings - it's harmonics on muted strings) while playing a note and while not playing a note, and the attack, duration, and release of the note.
For the first - it's hard to do (for me), but minimizing hand noise while changing position or fretting can enhance the "clean" sound. (I'm not good at this - but admittedly, it's not something I'm focusing on.) Additionally, being aware of sympathetic harmonics (especially if you have a very harmonically active bass, as I do) is another place to start. For example, I use floating thumb for muting, but due to my bass and string configuration all parts of the string are harmonically active, and the higher-order harmonics under my muting thumb sometimes ring sympathetically on the strings that I'm not playing (but actively muting). I've had to update my muting technique to address this, as I was getting low B-string resonance from harmonic nodes that's not noticeable at low volumes but comes through at gig volumes. (Embarrassing.)
For the second - I personally find that a clean attack isn't all that difficult, but a clean release can be more challenging. I learned to mute using guitar technique - reducing pressure on my fretting fingers to stop a note. This works on bass, but there's a noticeable difference between using that technique and stilling a string with your plucking fingers. (Again, harmonic nodes come into play here sometimes also.) It may not be as noticeable amplified, but for me, it's definitely noticeable acoustically - there's inevitably the tiniest fret buzz when I mute a note using the "liftoff" method instead of doing something different. I think this translates to a note release not being as clean as it could be in a studio environment.
Also for the second - you talked about note duration - that's very important, coupled with the release. I feel that I have some problems really "connecting" notes smoothly sometimes - like, I give each note maybe 95% of its value instead of the full 100%. For some styles this is great. For others, maybe not. It's just the way I play and I haven't gotten any complaints, but I'd like the play the way I do by choice. On the flip side, having a note duration that is too long is as bad if you are, for example, locking into a groove where the space is important (for example, leaving space for the snare in a hip-hop groove).
Some of this might not apply to you (hell, none of it might) but I understand where you're coming from and I can relate. Good luck - keep practicing.
