Realtors do list it correctly in their MLS systems. Unfortunately, Trulia does not have a separate system for co-ops and condos, therefore Trulia lumps both co-ops and condos together. Zillow never had separate categories either until recently. Zillow bought out Trulia in 2015 and since Zillow now owns it, Trulia has not updated the system since. So this means until the CEO of Zillow works on upgrading Trulia, forget about listings being separated from co-ops and condos.
As a guideline - the easiest way to tell the difference: If the asking prices are low (cheap), it's a co-op. If the asking prices are high (expensive), it's a condo. The reason is the form of ownership in a co-op is the buyer is a shareholder - legally not an owner. In a co-op you are buying shares of stock in a limited housing corporation that gives you "ownership" rights to "own" an apartment in the building. You do not receive a title or deed to your apartment. This is why co-ops are priced much less than condos.
In a condo - you are buying the apartment out right and you receive a deed or title. This explains why condos are priced much higher than co-ops.