Coffee is basically less concentrated espresso. And espresso is basically more concentrated coffee. (Well, a particular type of bean, but yeah) So a red eye is a cup of coffee with a shot of espresso to make it stronger.
So what you call a coffee we call a weak espresso. That no-one ever orders ever.
Er. not quite. I don't mean to sound patronizing, but there's some pretty serious misunderstandings going on here.
"Espresso" is coffee beans, very finely ground and then extracted at high pressure with steam about 15-20 grams at a time. Very tricky to get right and important to have the right equipment. You end up with a small shot (~1 oz/25 ml) that can be drunk by itself or mixed into other drinks. It's thick, syrupy, and contains lots of flavor and volatile oils and stuff when done right.
Espresso shot: you drink it straight out of the machine, default is a "double" but ordering a quad isn't unusual. My place serves shots with a glass of seltzer water but the big chain stores usually don't. You can buy it separately from their drink cooler.
Kinds of drinks on the American market vary a lot by region and by store. I go to a place that's very strict on its sizes... chain stores generally allow you to order small/medium/large of any drink, regardless of the format.
Macchiato: this term varies widely from shop to shop. my place serves a small cup with approximately 2 ounces of espresso and 2 ounces of steamed milk. Chain stores make this a big cup of hot milk with a little bit of espresso in it and add other syrups to it. A "caramel macchiato" is a bunch of steamed milk, several pumps of Torani syrup, and whipped cream on top with caramel sauce drizzled over the top of that--basically a milkshake with espresso thrown in
Cappucino: at my shop this is a 6 ounce drink with your espresso shot and the rest filled with steamed milk. Chain stores make this half-milk, half-foam with the espresso shot thrown in.
Latte: my place serves 10 oz drink with your espresso shot and the rest filled with steamed milk, at chain stores this is basically a Cappucino with no foam.
Americano: An espresso shot (double or quad) into which has been poured on top of a cup of very hot water (you pick the size). It gives the extra volatiles that come out of the espresso extraction process, but more diluted so you can savor it.
Edit: Reading further, the terminology is more precise: An americano usually is mixed. a "Long black" is espresso poured on top with the crema left intact. If you don't mix it, the espresso tends to stay on the top giving you a strength gradient as you drink it. My place doesn't distinguish and always serves a long black even if you order an americano.
"what we call Coffee" is more coarsely ground coffee beans, then extracted at (usually) ambient pressure with water just below boiling usually between 195-205F. You end up with 6-10 ounces (or way more) of more of a rounded or mellow flavored but still strong drink. Strength depends on which method of extraction you use and how much coffee you use. At home I use a french press, 2 oz/60 grams of coarsely ground coffee to about 20oz (600ml) of water and I let it steep for about 4 minutes.
Chain stores use a big drip machine with lots of coffee to brew it by the gallon. Quality and strength can vary widely depending on who's making it.
Most anything above can be served iced, particularly in the summertime. I don't usually find that an iced coffee or iced latte implies ice cream has been added.
Espresso over ice is sometimes forbidden because some people will then take it to the condiments bar and pour cream into it thereby getting a higher-priced drink for less money. Also, some purist shops say that the espresso is damaged by the ice and won't do it because they don't want you to mess up their coffee.
A redeye does sound like you've augmented a cup of coffee with espresso. You might want to do this at a chain store you knew had weak coffee or if you wanted to add the extra volatiles of an espresso pull to your coffee.
Espresso is higher in caffeine and flavor per ounce than coffee, but you usually end up drinking more coffee so your total serving ends up giving you more caffeine.
wow this post got long... does this help? anything else I can clear up?