Agree with the above. I would add that for that low point of draft, a collar would be a better fit than a breastcollar, since the line of draft will be putting a lot of pressure on the neck strap of the beastcollar. If you're going to keep the breastcollar, you'll need a vehicle with a high point of draft like a meadowbrook cart. With a breastcollar, the traces should continue in a straight line from the breastcollar - with a collar, you want the traces to exit the collar at as close to a 90 degree angle as possible. I don't really think a cart (2 wheels) vs. a carriage (4 wheels) is any better or worse for training - you can get into plenty of trouble in a cart too - but yes, if you're going to do a carriage, you really should get one with an undercut.
I'm particularly fond of marathon carriages and if I ever get back into driving that will be the vehicle I go with. They're sturdy, you can bring passengers, they're up to off-roading if you like - all around great vehicles. A nice phaeton is pretty sharp looking though. I would also highly recommend you spend the money to buy a vehicle with brakes, whether you get a cart or a carriage. Brakes will not stop a horse bent on bolting, but they will make bolting a bit harder for the horse. Brakes also are exceptionally handy for keeping the vehicle still while stopped at a traffic light, or for keeping the full weight of the vehicle off the breeching when travelling downhill, making it easier for the horse to balance.
I really like Frey carriages (https://freycarriageshop.com/
), and they can be custom ordered to whatever you need, but they are fairly spendy. They are absolutely top of the line though, and have impressive customer service. Roberts Carriages also makes nice stuff (Buggies & Wagonettes
) and are significantly less than Frey's. I really like their wagonnettes, which are basically marathon carriages.
I would really caution you against buying anything used, anything cheap, or anything handmade. This is one instance where handmade is NOT better. You are putting your life, your horse's life, the lives of bystanders, and your horse's trust at stake when you hitch to any vehicle, so make sure the vehicle is in good condition and won't break right when you need it least. Unfortunately if you want a decent cart you're going to be shelling out 2 grand at least for an entry level one, and upwards of 4 or 5 grand for a decent carriage. Driving is expensive, there's no getting around it, but you get what you pay for. If you can't afford something decent at the moment, don't drive your horse until you've had the opportunity to save up and get something good. Trust me on this.
Barry Hook has a channel on YouTube and he trains horses to drive beautifully. He's got some great videos up if you peruse his stuff, great training tips, great ideas for desensitizing, stuff like that. I highly recommend you give his videos a look.