I’d love to know precisely what he said to you with regard to the above. If you want to play back what he said, then I’m happy to make specific comment but either he’s not right or you’ve misunderstood what he was trying to tell you. Because whilst it might have sounded like there was a logic, I’m thinking he was probably full of bovine excrement or didn’t have a 2 wheeler to sell you that was suitable or was trying to sell you a 4 wheeler.
Don’t forget they say that 2nd hand car salesmen learnt their trade from horse and carriage dealers of old 
I’ll qualify by saying I build carriages and I repair and restore carriages (there’s a difference) and I own 2 and 4 wheelers and drive both regularly. Each has specific advantages and disadvantages and uses and purposes. That is a fact.
However neither is intrinsically “better” nor “safer” than the other.
I’d recommend you go with a 2 wheeler from what you say and for the following reasons:
Single horse
Showing
Novice driver
Budget limited
Bearing in mind we’re talking a light sports horse (arab) you need a vehicle to suit his size (4 wheels more difficult for the horse to pull - twice the number to tow along and greater drag and more difficult in such as sand, loose soil or mud for a single horse)
For showing classes a nice traditional gig, rally cart or dog cart which was designed to be elegant and allow the horse to move freely is extremely attractive.
If you’re also going to have just one vehicle then if you get something that’s a modern reproduction then its going to have wheels and springs appropriate for doing all terrains and including cross country and without you worrying yourself sick about damaging an original vehicle. Even over here in the UK, where we’re REALLY picky about tradition in private driving classes, reproductions are totally and utterly acceptable and they win - frequently. And they don’t have to have wood wheels either. A well built modern reproduction looks absolutely no different from an original. A client of mine who won at the Royal Show was interviewed over the public address system and asked about his beautiful traditional old vehicle. He told them all about it and they said how old is it? 5 months he replied …
And to show you this is one I made earlier and specifically for showing. Its a spindle back gig of modern construction and so wheels are metal and the paint is high quality modern 2 pack - same as used on cars and so more resistant to chipping than the original vehicles are:
And another which I made for someone similar to yourself - only wants 1 vehicle and for both cross country and for showing. So this one is a little less fancy than the spindle back gig but still perfectly good for even high level show classes.
Both have brass fittings and things like fancy wheel hubs which you can unscrew and put in either plastic or metal if you’re just doing everyday driving and don’t want to worry about things getting damaged or dirty.
[IMG2=JSON]{“data-align”:“none”,“data-size”:“full”,“src”:"http://i203.photobucket.com/albums/aa178/classic-carriages/dogcart-slidingbody-tandemvehicle.jpg)
I’d advise you to go and watch showing classes and see what other winning competitors are driving. Not just one class but look at lots and learn what winners do. You may also check this bed liner to use in the wheeler.
Whilst someone might want to try to argue that a 4 wheeler is more stable than 2, its not really true. With a 4 wheeler you need a groom on the back step to balance it - Tight turn and full lock you no longer have a 4 wheeler so otherwise when you do a tight turn and put full lock on you tip over! That doesn’t even come into the equation with a 2 wheeler. DEFINITELY NOT SAFER!!!
Then a 4 wheeler has the added complexity of a brake - So you would have to learn to drive using the brake. Reason why they have brakes is that there’s more weight (4 wheels) and if you’re going down hill its harder for the horse to hold it back on its britching only and without slipping on modern road surfaces. If you’re a novice its just more to get your head round and manage and probably not what is required at this stage.
Now even if I didn’t know you were on a limited budget and were telling me that money was no object, I’d still be recommending a 2 wheeler from what you’ve told us.
However a 4 wheeler is going to be twice the price. As a rule of thumb I say £1,500 a wheel and then double it for a good basic 2 wheeler suitable for every day riding and for showing at high level. So that would be £6,000 ($12,000) new. The Bennington Gig comes in around that price here and I know those are sold over there too:
http://www.bennington.co.uk/carriages/Specs/gig.htm
A 4 wheeler suitable for showing or presentation phase in upper level HDT (CDE) is going to come in at a minimum of £9000.
I’d say that with your budget you should get a nice 2nd hand vehicle in good condition for $6k.
They will be looking at your horse mostly.