Peggy-
Your idea is correct, and actually, that buggy does have a kind of 'cut under'; what it means is that the body( of a 4 wheeled vehicle) has some sort of 'cut out'-usually looks like an upside down, square-cornered, "U", that is usually located about under the driver's seat. It occurs in combination with some sort of 'fifth wheel' arrangement which allows the front axle to rotate(there would be no use to it if this weren't the case!)further than it otherwise could. I can't tell from the photo if the 'surrey' under discussion here has that or not? It is possible that because the design of this vehicle appears to be basically taken from that of the EE pipe carts, that it has a 'cut under' only because it is basically, two EE carts placed together-but if it does have a 'fifth wheel'of some sort, then of course, it would be more manueverable and the design would actually SERVE AS a cut-under, and not just 'appear to be' one...
Nowadays, there are all SORTS of 'fancy' innovations, such as delayed steering, extendable axles, etc. on MARATHON vehicles for CDE competition and the like; however, such things aren't yet widely(if at all; I haven't seen evidence of any, at least) available in miniature horse size, so no need to be considered in this post--So,speaking of more 'basic' horse drawn vehicles, then---the most manueuverable 4 wheeled vehicles (meaning basically, that they can make the TIGHTEST turns !)generally are those WITH a fifth wheel front axle arrangement, AND, WITHOUT a 'reach'-which is basically, a connection straight down the middle of the undercarriage that joins the back axle to the front one. A reach would 'strengthen' the overall integrity of the undercarriage, BUT, it also can limit how far the front axle can rotate, and thus, the maneuverability of the vehicle-whether the vehicle is cut under or not! It is part of correct design for any vehicle with a reach, or where in a turn, the wheels would contact either the body of the vehicle OR the reach, to have 'rollers' mounted on the reach/vehicle body at the points of contact, for safety, so that the wheel would not be 'braked' by such contact-which would be DANGEROUS and cause a wreck/turnover!
My most manueverable 4 wheeled vehicle is my Glinkowski mini marathon vehicle--not surprisingly, as it has no reach,is cut under, and has a fifth wheel. Both of my others have a reach(both with rollers at the wheel contact points)-the one w/ a cut under,and much larger wheels, rolls more easily, and has a better turn radius, than the one( a little buckboard, which I LOVE for parades and the like!) w/ small front wheels that can pass UNDER the wagon box--but NEITHER of them would be a suitable vehicle for marathon driving.
As you can see, there is a lot more to this than first meets the eye!
Some years ago, I saw and LOVED the look of, a mini 'covered wagon' being built by some folks in the midwest, at AMHA Nationals....however, when I saw it 'in person, I could see(and as was pointed out to me as I stopped him and discussed the vehicle,by the longtime driver and AMHA judge who was driving it around for an little exhibition), it was NOT a practical nor particularly safe vehicle, because it had almost NO turn radius, and would have toppled over if taken at any speed into any sort of tight turn....
Sorry-I've been verbose as usual--but hope this answers your inquiry and gives a bit better understanding of the cut under and how it works!
Margo