A Fazer does both fog and haze. You simply run it at a lower output to get a fazer to haze. All fazers (that I know of) are water based only.
Most hazers are oil based. There are some that are water and oil based. The HZ-400 does both oil and water based for example. I am still waiting on an answer to my question with regards to the HZ300 and whether or not it can use water as well as oil based, as seen here: http://forums.elationlighting....31084274/m/745105374
Oil based tends to hang longer and look nicer then water based, but creates problems with residue and some don't like the smell. Oil is also more likely to trip beam sensors and smoke sensors then water based.
Hopefully this helps and answers a few questions.