Talk-a-Round simply means "talking around the repeater". So, Talk-a-Round (aka Direct) is where the mobile or portable talks simplex to others on the output frequency of the repeater they normally use. It is primarily used when a repeater is out of service, or you are out of range of it.
Unfortunately the meaning is skewed now days. With all of the trunking systems, Talk-a-Round has has been commandeered into describing conventional repeater channels that are stand-alone from a trunking system, and also to describe car-to-car type talk groups on a system. It has been my experience that very few 700/800/900 systems have any actual simplex "Talk-a-Round" capabilities. Which in my opinion is a shame. That means in a failure, you can't even talk to the guy standing 100 feet from you on that fancy radio.
Trunking systems talk groups should be labeled as car-to-car, field ops, chatter, non-dispatch, etc. And, stand-alone conventional repeaters should be labeled as back-up, etc. But, the way things are, most people using radios now have no idea what Talk-a-Round really is.
Our local Sheriff's department has cars that go to "direct" (which IS true Talk-a-Round in this case, they run a conventional simulcast system) but they don't understand why the dispatcher is talking over them, and doesn't know it. They have a car-to-car channel and never use it. Goes back to training and operator knowledge.
I'm sure someone will disagree with me here. And, I realize the OP was in relation to a specific system in a specific area the OP lives in, but the topic was general and I was in the mood.