I would say the budget would be around $400 - $500.
I'm not sure about the second question by accurate do you mean distortion vs. clear? I'm primarily a classical player so don't need a thumping bass.
Accurate means just that. Correct reproduction of the source. An example of not accurate is thumping bass. When you listen to really good audio. You go back to other and am surprised at how innacurate they are. You don't realize until you hear the really good audio.
Many of the popular monitors at a good price have inaccurate bass. The reason is that people like that. Although I haven't bothered to listen to many in your price range. I imagine they aren't real good.
I have two pairs of monitors for my audio.
One pair of Event Studio Precision 8.
One pair of Prodipe Pro Ribbon 8.
The Event's are really good monitors. Fine reproduction. Not harsh. Very accurate. I'm happy with the bass. I live in an apartment. I can't go using a sub woofer. These do have one achilles heel. They will burn out the amp. These are not sold anymore. The remainder I've seen online. They are asking $1400 a pair for them. A newer version with a different name is cheaper. But a little above what you are wanting to spend. Event 20/20 bass
They look much like mine. They have a heat sink on the back. This indicates to me this is an upgrade to eliminate problems with the amp. I would suggest these. Event does an exceptional job on sound. Mine, I will keep. I'm happy with them even though I many have an amp in one of them replaced every so often. That costs me $350. I'm still happy enough with the sound to keep them.
The Prodipes are a budget monitor. The company is known to put out a lot of junk. In the midst of this they will turn out a diamond every so often. Mine cost $350 new for the pair. I haven't had many of the problems others have experienced with theirs. Like hiss from the speaker with nothing on. Or an led not working. Mine were very tight at first. Had had to cook them a bunch. They really aren't bad at all. A little on the hard side. I compare a good source through them and my Event's. I love my Event's. They're just a big step up. Finer, more accurate, nuances right. I love my Event's.
Overall I won't criticise my sound. I'm running the same PreSonus sound box many here do. It isn't that good. There are much better sound boxes. I'm running IvoryII for my piano. I have a pair of AKG K240 headphones. I'd put my Event's on par with the headphones. You can get better. Both of these are good enough...they meet my baseline for good audio.
I wouldn't get anything MKII. That usually indicates the increased drawn out false bass.
I would like to add. The other day I played a couple Mariah Carey songs off youtube through my Event's. Geesh! When they are fed the source. They will hit you hard with the bass. But it's accurate. Not the false drawn out bass on many of the popular inexpensive monitors these days.