Ok, first thing to know is, anything with shanks is considered a curb bit, even if it has a broken mouthpiece. Bits are defined by the type of action they use. If it has shanks it's working off of leverage and therefore it's a curb bit. Snaffles work off direct pressure and can also have solid mouth pieces, some even have ports.
No you don't need a chin strap with a D ring. Some people will put them on an O-ring to keep the bit from sliding through the horse's mouth, but It's not really necessary with the D. The purpose of using the chin (sometimes called curb) strap with a shanked bit comes back to the leverage action of the bit. When you engage the curb bit, the strap puts pressure on the horse's jaw and pole.
If you are just trail riding, there's no reason to ride in a curb unless your horse is responding better to it. The bit you're describing as a shanked snaffle sounds an awful lot like a tom thumb. There have been several threads discussing that bit and why its typically not the best one to use. They have some great information on curb bits in them if you want to check them out.