You'll get whatever model they send out - one of the HR2x series. It may be new or a refurb. There's no way to specify what you get. If you want to record OTA (gives you additional recording options), you need to ask for an AM-21 OTA add-on box. Normal price for the AM-21 is $50 extra, but you can often talk them into sending you one if you're out of contract and threaten to switch to Dish or cable.
There are no Tivo brand DVRs available for DirecTV at the moment. They have decided to partner with Tivo again, so they should be available again in the next year or two, as a premium product/service, in addition to their current DVR line.