Making recordings better? Preamp/DI box question.

Hello!
The story: I use Fender Jazz Bass passive directly to M-audio Fast Track, which goes to PC or MacBook with Reaper/Logic, but I don't like the quality of the sound, it is kinda plastic and cheap. Even with VST plugins. I see many youtubers get really clean and rich sound, using preamps and DI boxes (I don't speak about those guys using big amps and mics), and I'd like to make my sound better.
The question: what should I get to be able to record at home with good quality? A DI box, or a preamp such as tech 21 sansamp bass driver di programmable, or shiftline olympic mk2/mk3?
And 21 sansamp can be used as DI box, so is it best from both worlds?
Also I have a fuzz pedal bigmuff deluxe, which also can be used as DI box, can it help me improve my overall sound quality when recording?
And yes, I don't use old strings.

Hope you can give me good advice, I'm quite new to this and really need help. Thanks!

 

Thanks!
About the old strings — anywhere I asked the first answer was like "dude just put some new strings" which of course wasn't that helpful.
RJB and REDDI are out of my budget for now. I just want a simple home set for playing. In fact I AM playing, but I really want to make sound better, without sacrificing a lot of money.
What if I change M-audio Fast Track to Focusrite 2i2, for example? I don't think it gives me more clear sound, but I just wonder.
Also I heard a lot that SansAmp BDDI is worse choice than sansamp bass driver di programmable.
I can record what I'm getting now if you're interested and give some examples of what I want to reach.

 

That's the amount of money I may spent. But will it make difference? I really think that without better pre-amplification the better sound card won't make any sense (maybe just a little).

Also a very stupid question: on that picture there is a top "head" amp — the SansAmp and ShiftLine preamp pedals do the same job but in much compact pedal form?

 

Mark Studio 2, by Overloud.

 

I'll record something for you to listen. Maybe I'm too picky.

 

That is with mark studio:
This is the same, but without any effects, absolutely clean:
First of all: sorry for mistakes, I usually do better, but right now I'm little bit sick with temperature (I know it is sounds like justification, but who cares?). Think it is enough to check the sound.
Second: I know I could get better if I use mic input on my card and use bass amp to record it. But usually I play late at night, so I can't make much noise and my only option is to get better sound quality without recording an amp. Only bass to sound card (maybe with preamp, I still not sure).
The settings of MarkBass are those:

 

The input on Fast Track is almost maxed out. A little bit more and I get clipping.
In reaper and markbass there's optimal settings, but the sound still quiet.

 

I didn't quite understand. What do you mean, can you explain?

 

I still don't get it. What "guitar button". It is in Reaper, or in VST? I haven't seen such setting.

 

Ah. No, I don't have options there. As I mentioned earlier I use M-Audio Fast Track.
For the sound reference I can use this video
As I see they use bass guitar to Markbass Little Mark Tube 800 to Apogee Sound Card to MacBook with no VST plugins.
So what if I stay with Fast Track sound card (I really don't think that the sound card makes that much sense. other than loudness and noise amount) and use markbass preamp, or any other (shiftline for example)? Should I get better sound quality? Also preamp can be helpful on stage, so I get x2 benefits.