Your assumption is correct—it's both!
When there's a chance of exposure to filth fever that's not dependent upon a creature exposing an adventurer to the disease, the disease filth fever typically requires succeeding on a Fortitude saving throw (DC 12) to avoid infection. That saving throw DC is used if the DM determines that the surroundings are so filthy that filth fever is present: "Those injured while in filthy surroundings might also catch it [i.e. filth fever]" (Dungeon Master's Guide 292).
However, the extraordinary ability disease of the dire rat (Monster Manual 64) has its saving throw DC computed like most creatures' special abilities are. The Monster Manual on Special Attacks and Special Qualities, in part, says
When a special ability allows a saving throw, the kind of save and the save DC is noted in the descriptive text. Most saving throws against special abilities have DCs calculated as follows: 10 + 1/2 the attacker’s racial Hit Dice + the relevant ability modifier. The save DC is given in the creature’s description along with the ability on which the DC is based. (6)
Thus the saving throw DC against the dire rat's extraordinary ability disease is 10 base then +0 for the dire rat's one Hit Die (rounded down) then +1 for the dire rat's Constitution modifier for DC 11 total.
While this does make the disease filth fever more dangerous to adventurers when it's present in the environment (Fort DC 12) than when it's carried by the typical dire rat (Fort DC 11), most of the time the environment isn't also trying to eat the adventurers. Most of the time.