I do daycare, and get my kids on the big potty ASAP (from the get go if I can), but some of these just turned 2 year olds are freaky with the big potty. If you go for a stand alone, my fave is the Fisher Price Froggy Potty (about $12 at Target or Walmart..2 pieces, formed nicely, no sneaky spots for leaks, etc), or just get an IKEA one if you have an IKEA nearby (one piece, cheap..under $10 I think).
Another option at home is the Potty Pal.
http://www.pottytrainingconcepts.com/Flip-Toilet-Seat-DRE...
Its toilet seat for oval or round bowls....and it has a 3rd insert to your seat..its part of the flip up lids. There is the regular seat, a thin seat reducer for little bodies, then the lid. I understand that out in public we need our kids to be flexible with potty arrangements...but at home teaching independence with it might be better, or at least something to consider. If they can go easier by themselves (with something like the Potty Pal) and for you isn't any extra clean up but one more seat-layer to wipe down with that cleaning disposable wipe equaling a few more seconds of your time...might be worth it.