As other people mentioned, it does vary by state whether CNMs can attend homebirths or not. In my state (MA), they cannot, so homebirths are attended by CPMs or direct-entry lay-midwives.
A CNM has a nursing degree in addition to midwifery training, and they usually work in hospitals. Generally, they are less interventionist than doctors, but sometimes they can find their hands tied by policies (risking out certain woman, postdates etc). They may also have a more "medical" model. A CNM does not go to medical school, however.
A CPM is also a certified midwife who is licensed and certified by NARM. In CO, I believe HB midwives cannot suture, but a CPM is trained and qualified for most birth emergencies. A CPM attending a homebirth would (should) come with emergency supplies, and usually a backup midwife. These include oxygen, neonatal resuscitation (deLee suction), as well as drugs. They would not induce labor with pitocin, but if you were hemorrhaging after birth, they would give you pitocin or methergyn to stop the hemorrhage. The reason you'd likely have 2 is that one can focus on the mother and one can focus on the baby. A CPM is a good option for a healthy low-risk woman who choses to have a homebirth. A good CPM will pre-screen her clients, and over the course of the pregnancy develop a relationship with you. You should ask about transfer rates, experience, philosophies (i.e. cutting cord immediately vs. waiting until Placenta is out). NDFanatik had a great post on questions to ask a midwife. While the majority of transfers to a hospital are non-emergency, in most states a CPM might not have backup doctors (this is due to political reasons, because in many states midwifery is a-legal, meaning it is licensed, but unregulated), so you would have to just show up at the hospital in the even to transfer.
I would definitely meet with different kinds of midwives. It all depends on what YOU are comfortable with too. Depending on whether you would prefer having a hospital next door vs giving birth in your own home, both a CPM or a CNM could be an excellent fit for you. I think the most important thing is having a care provider that YOU are comfortable with. If you have a bad vibe, trust it, and if you really click with someone, that is important too.
Just so my biases are upfront, I'm training to be a homebirth CPM, so obviously I fully support what CPMs do! I'm also planning on having a homebirth CPM attend our birth. In the event we need to transfer, I will have to likely just go to the hospital with whatever doctor is on-call. I don't think that a homebirth is for everyone, and it is not my place to judge what is right for anyone else. However, I encourage you to learn more and ask the midwife all of these questions- she will be able to provide you peace of mind more than anyone else here can! No matter what you decide, I hope you feel inspired and empowered, and find someone you totally connect with to attend your birth!
P.S. Pseudo-diva- I'm sorry you had such a negative experience at your HB- that sounds totally unprofessional. :(