All I can say is that every source I have accessed from the wider world of driving indicates that the whip is indeed to be held in the right hand, and at an approximate 45 degree angle, both 'across' the driver's body AND, relative to the line of the horse and cart -hard to describe, but easy to see.( I have never come across any reference to a 'one o'clock' position.)This puts the whip in a more 'logical', if you will, position for use when required, yet it isn't in your, or any passing driver's, way. For a single miniature, the lash should not be all that long-probably 12 to a max. of around 16" would do(just my guess, based on what I have learned)-a whip with a longer lash is made for a multiple hitch, and the 'longer' the hitch, generally, the longer the lash would be, within reason. I am far from as proficient as I would like to be; however, I have done a fair amount of practice with a lash, working on learning to 'roll' it to make (gentle)contact. A tip--it can be much more comfortable to hold nearly any, and especially, one of the more basic(i.e., inexpensive, mass-produced)whips if you hold it with the heel of your hand just on the top of the handle, or 'butt'. When held at a proper angle, such a whip can become tiring to your wrist/forearm(not really a problem if you are holding it so that it sticks straight up, but --this isn't the proper way!)--holding it as I have described helps immensely with the balance. To do this, you need to use a slightly longer whip that you otherwise might think. Remember when taking the recommended measurement, that it says to measure from your HAND....not from the far end of the handle or 'butt' of the whip! (and be sure you are not reaching forward when you measure, but in a natural but upright seated postition, with upper arms parallel to your torso, and forearms approximately perpendicular to upper arms, in a naturally comfortable position.(One of the drawbacks, IMO, of miniature carts-especially show carts-is that the driver is seated so low relative to the horse that it is hard to keep the arms in a correct position to hold the reins properly. A dash helps somewhat, to keep the reins up off the horse's rear, without the driver ending up with upper arms up and out from the body, and forearms angled upward-just look at many show photos......)
I think many non, or new-to-it, drivers, may visualize "whipping"in their mind's eye when they even hear the term "whip"-when in truth, seldom if even does one actually have cause to apply the shaft in a strong contact, and then only under drastic circumstances-at least, IMO. To me, the aim should be to have the horse learn to accept the contact-and in training, it may need to learn to connect a verbal cue to a follow-up of a 'tap' on top of the rump or around the area of the harness saddle, as an impetus to further impulsion-and also, to learn to yield to contact (on one side of their torso or the other-again,usually in the general area of the saddle.)
The 'ideal' is to drive one-handed on the reins(Achenbach-style), with the whip then by itself in the right hand, so that any use of it in no way affects the reins. This is a lofty goal, however!
One can, with practice, though, learn to use the whip held 'on top of' the right rein, in a manner that works, and has minimal effect on the horse's mouth through movement of the reins.
The popper is generally rather 'stiff', while the lash is somewhat less 'rigid', while not being really a 'soft lay'("lay" is a term we used to use with lariat and other types of rope). If the lash is too 'soft'(or, as MiLo said, too long for the hitch it is being used with), it may well be prone to tangle. I believe the "popping noise" only actually 'happens', at least to a noticable degree, when using a whip with a longer lash; you likely won't notice much of a sound with a short lash or popper-only, entry-level whip-but, it is the traditional way to 'finish' the whip. I have seen whips with no real lash, just a popper, described as "buggy whips", FWIW.