The accurate DOM count is subjective according to who you ask. If you were to look at the rules of one MLS vs. another, we would find different criteria for counting DOM. For Trulia, the count may reflect how many days it has been active on Trulia site. Perhaps Trulia can comment on this.......
In some MLSs, if the listing expires, and 2 days later, it is reslisted by the same agent/broker, the clock is reset. In another MLS, the clock is not reset, but days are continuous, even with the expiration.
You mentioned that the Realtors pass this lisitng around like playing cards. The guidelines for counting DOM in your market may reset when the property is relisted with a new broker/agent.
If the DOM is intentionally being misrepresented, that action should be reported. If the change of brokers reset the counter, that might be consistent with the reporting rules for that area.
For detailed accurate info on a property, consult a Realtor. As more and more sources for data on properties emerge, discrepancies will increase. The explosion of real estate data brings many benefits to the consumer and some challenges.