Is a water pump part of the engine or is it considered an accessory?

It is part of the engine but it's considered "an engine-driven accessory". It's true you do need it to drive the car, but the same can be said of the gas tank. Most extended warranties, which are just over-priced service contracts, cover "internally-lubricated" engine parts meaning anything that's in contact with engine oil. The better contracts spell out whether things like water pumps, intake manifolds, and other bolted-on stuff is covered or not. Some contracts put that in the really fine print to surprise you later. Too many times I overheard at the dealership, "what you need isn't covered". Your best recourse then is to cash in the remaining value of the contract and use that money to pay for the repairs.

As a general rule, whether it's tvs or cars, service contracts are not a good deal for most people. The original warranty will cover the bugs that show up on every car when it's fairly new. Cars today DO have a huge pile of electrical problems due to all of the insane, unnecessary computers to do simple things, but engines and transmissions can easily last 200,000 miles if they're maintained and timing belts are replaced on schedule, when they're used. Many expensive repairs cost less than the cost of service contracts so you're better off just putting that extra money in the bank in case you do have a big repair later.