Would you recommend that I keep the cutting in water longer?
My feeling is that you should get it in soil before the roots elongate too much, an inch long seems like it should be OK. And by soil I don't actually mean 'soil' at all, see below.
If I buy some potting soil, is there a special bonsai mix?
Bonsai soil is a subject that could easily take a whole chapter in a book. Please read the sticky thread about soil and re-potting. Once you have a general idea we can elaborate on any parts that are unclear. In general bonsai soil appears rather coarse to someone who is accustomed to seeing potted plants in the usual commercial potting soil. What is essentially small rocks are used to ensure drainage and aeration.
Do you have any other plant supplies on hand? I have found Perlite, that is readily available an nearly any garden center, to be a good component for young material. Brent (EverGreen Garden Works) use 50/50 Perlite and bark for his plants. And so with you, a cutting does not necessarily have to be in premium material right off the bat but drainage is essential. I have taken to using a coarse, bonsai type, mix for all potted plants, they seem much happier.
Make sure to follow the links in the sticky and check out the section on using kitty litter\oil-dri at Bondsai4me. The link should be in the sticky. Of course ready made bonsai soils are available as well but you end up paying to ship rocks. I think a re-potting is probably in order for your older plant so whether you purchase or mix your own medium for it you should have a little left over for the cutting.
P.S. Sorry, this post was not as well organized as I would prefer, a little short on time right now. Happy to elaborate on anything that is unclear.