Both contractors are calling it a drip edge but they are two different things.
The roofer's drip edge is a metal edge that you could call flashing. It's L shaped about 8x1 alu. and is attached along the roof edge before the shingles. It covers and protects the exposed edge of the roof sheathing and the top edge of the facia board, it also repells water from dripping down the facia board.
The carpenter's 1x2 and calling it a drip edge is also known as a shadow board does a couple of things. It adds a bit of detail to the facia and will also hold the gutters away from the facia board which helps to prevent rotting of the facia.
The roofers metal drip edge should extend over the carpenter's "drip edge" which needs to be installed before the roofing. The carpenter should rip cut the top edge of both boards to match the pitch of your roof.
You should have them both installed and the painter should caulk and paint before the gutters are installed.
You may also want to consider some of the newer pvc type trim stock that will never rot.

Be sure to use 5/4 wood for the fascia.
1x warps and does not hold fasteners for the gutters.
Any end grain can be treated with Minwax wood hardener to limit moisture uptake, end grain is like a pack of straws just waiting to suck up water.
Paint does not form a very good barrier here since the resin is absorbed into the wood and leaves a poor film behind.

Thank you beacher, I kinda understood what the carpenter was trying to explain to me, but I was still unsure. You have made it much clearer for me. Thanks for the tip on the gutters as well, it makes sense to caulk and paint before reintalling the gutters. I was not aware of the pvc type trim stock, but then again, this is a whole new adventure for me. When I spoke with my carpenter, I explained that we would most likely be selling in the next 3 -5 years and wanted a good middle of the road type material. We are still paying on major foundation work from last year and while we want this job done right, we can't afford to go high end materials. He suggested masonite. So that is the plan. Is there any reason why that would be a bad choice? And as I type this I realize that I don't know if the trim stock and facia wood are the same thing. Sorry to show such ignorance, but then again, that's why I'm here. Great bunch of people over here to help find my way through this project.
Brickeyee, thanks for the suggestion of the minwax on the end grain. I am guessing that the end grain is the acutal end of the piece of wood? And does the paint go over the minwax with any problem?
Again, thanks so much...I really appreciate the insight and help. :-)
Erin

I've never used masonite outside so I can't answer that. You could just call around and get pricing on the three products, masonite, pvc and primed pine. Facia is usually a 1x8 but yours may be different.
As for 5/4 stock [1" thick], I wouldn't use it. The facia is nailed 16" on center and adding the 1x2" drip edge will give you an 1-1/2" thickness for the gutters.
Also your end grain, yes, the end of the board, should be butting up against the rake boards. The rake boards are the trim boards on the gable end of your roof.
The facia trim is horizontal on the front and back and the rake trim angles down from the peaks on the ends.
As long as any exposed to the weather end of your trim is primed and painted, that's all you'll need.

Beacher, thanks again for helping me understand better what I need to know and clarifying terminology as well. I appreciate all of your help.
Erin

In your original post you mentioned trim craft which is a brand name for an engineered wood trim product. There are some pictures in the link that might help understand how everything fits together.

Here is a link that might be useful: TrimCraft Cornice System

You do not place masonite on the exterior of the house. That was a stupid, stupid suggestion from the carpenter.
Ron

Would you care to expand upon that Ron?