Hi Joy,
About 6 months ago I was in the same dilemma as you going back and forth between these two items on One Step Ahead's website! I thought the idea that they folded up was great (so compact to fit in a diaper bag), but was discouraged by some of the reviews for both that said they don't fit well on elongated toilet seats. I mean honestly, the whole reason I would be buying it was so I could use it out of the house and the majority of public restrooms have elongated toilet seats, so if it didn't fit well on there, then what was the point of getting one? So I ended up changing my mind about them and ended up doing more research online. I ultimately went with another product called the "Potette Plus 2-in-1 potty" by Kalencom. I had a good friend who had one and loved it and honestly I think it's the greatest and wouldn't buy another product again. It is a little bit larger than the folding potty seats, but it is a versatile product in that it can be its own free-standing potty (in a pinch) or can be used as a potty seat on top of a regular toilet. I use it mainly in public restrooms (that's what I really was looking for), but the free-standing potty part has come in super-handy when we are somewhere there are no restrooms (certain parks, long car trips where there are no rest stops, etc.) or someplace where I can't just quickly run in and take my daughter to the restroom. For example the other day my son was asleep in the carseat while we were driving somewhere and my daughter all of a sudden had to go really bad. So rather than find a place where I would have had to lug her and my son out of the car into a public restroom, voila! I just pulled into an empty parking lot, set up the potty right next to the car, she did her business, and we were good to go again. My son slept through the whole thing!
Basically how it works is that it is a potty ring that has folding legs that can make it stand upright as a free-standing potty or fold completely out to the side to support it on top of a regular toilet. As a free-standing potty it comes with these plastic bags that have a very absorbent pad in the bottom of them that are biodegradeable so you can just toss them when done. This potty seat is smaller than the other potty training seats you can buy that fit on the top of regular toilets too. It comes with it's own plastic storage bag to tote it around. Originally I was worried about carrying it around with me because I thought it might still be too big, but I found it fits easily in the open side pocket of my regular-sized diaper bag. So no problems there.
Here is a link to the exact one (red) that I bought off Amazon: http://www.amazon.com/gp/product/B0016L0MMS/
They have other colors too and you can read some of the reviews if you want. You will have to buy the refill bags separately though if you need them. The potty comes with 2 refills when you buy it, but I bought a pack of 10 refills just in case. Although I mostly use mine as a toilet seat in public restrooms, (I have only used like 3 of the bags so far), it's great to have them when your child needs to go right away or for the convienence!
Anyhow good luck choosing and good luck potty-training!
M.
P.S. One final note, if you do decide to go with this product, make sure you buy the Potette Plus product and not the original Potette on the go potty. The Potette Plus is the one that has the ability to be the free-standing potty plus the toilet seat, while the original potette is just a free-standing potty.