After recently moving into our new house, we find our (wood burning) fireplace chimney has no damper. So we had a few chimney people to give us quotes and there seem to be contradicting views on whether to install the newer type top damper, or the traditional bottom damper.

Here's the benefits that I can make out:

Top Damper:
Has a silicon/rubber seal so will be very efficient.
Cheaper, but likely to deteriate quicker.
Quoted $525 including installation

Bottom Damper:
Made from cast iron so won't form a perfect seal.
Will last a lifetime.
Quoted $800 inclduing installation.

We do already have a good quality glass front door which I'm hoping does reduce air escape well.

What would you all suggest, and are these prices reasonable?

The main thing you need is a cap if you don't already have one to keep the rain out. If that's the case, I'd lean toward a chimney top damper. I put in a Chimalator, and a quick check of prices shows they've escalated to $200+ for the kit from the ~$100 I paid for ours. Price depends on the size.

Honestly, I think the pricing you got is a bit steep for either type of damper, unless you have a massive fireplace and flue and a roof so steep that it'll be a giant pain to get to the top of the chimney.

For the chimney top one, they're going to go up on the roof and put the cap/damper in place with screws and some RTV silicone as a gasket. They'll drop the stainless wire that is used to close the damper down the flue. At the fireplace, they'll drill holes in the masonry for the bracket to screw into the wall. Then they'll attach the wire from above and set up the tension so that the damper is closed all the way when the lever is all the way down.

That's maybe a 90 minute job for someone who knows what they're doing. Marking up the material by 2X to install may be pretty normal. It's also why I DIY most things.