The short, strong, thimbleful of coffee is a café solo.
The short, strong coffee with a drop of warm milk is a café cortado.
The short strong coffee diluted with about the same amount of hot milk is a café con leche.
The short, strong coffee, diluted with hot water is a café solo largo (but, as foreigners, we can say café americano).
The name americano, for what Spaniards traditionally call café solo largo, for weak black coffee, is the Italian influence. Most Spaniards think we foreigners are strange for watering down good coffee so they are not too concerned about what we call it. It's just another odd thing we foreigners do. Americano never has milk. We are not in Starbucks or Costa Coffee with their funny ways; bucket sized cups and exorbitant prices.
The beans are usually ground alongside the hissing coffee machine. Grinding beans is noisy too. On very rare occasions you may be offered different beans. I can't remember the last time that happened but I have a pal who always asks if the beans are torrefacto, that's roasted with sugar, because she doesn't like torrefacto. They never are. Everywhere, well nearly everywhere, offers decaf beans nowadays. If you want decaf from the hissy machine you have to add descafeinado de máquina to your order - ponme un café solo descafeinado, de máquina, por favor or me pones or un café con leche descafeinado de máquina, por favor etc.
There are tens and tens of variations which are to do with the presentation and the amounts of water, coffee and milk. Here are just a few. Lots of people ask for their coffee in a glass (en vaso). Foreigners sometimes ask for a big cup (en tazón). Corto de leche means go steady on the milk and corto de cafe means easy on the coffee. If you want a strong coffee you can add, bien cargado de café. I understand that, in the modern world, lots of people don't care for cow's milk so you might need things like leche de soja (soy milk) leche de almendra (almond milk). If cows aren't the problem but fat is you might ask for leche desnatada (skimmed milk). If you want sweetener instead of sugar, which will come by default in sachets, ask for sacarina. Nobody asks for brown sugar. In Summer people pour their coffee over ice. Just add con hielo to your order.
The short, strong thimbleful of coffee poured into a glass which already contains condensed milk so that it looks like an upside down Guinness is a bombón. This may be a Costa Blanca and Murcia thing.
The short, strong thimbleful of coffee poured into a glass which already contains condensed milk so that it looks like an upside down Guinness and then with a splash of booze added is a Belmonte. Again specify the alcohol of your choice. This too may be something local to this area.
And finally an easy way to make Spaniards snigger, as you sip your coffee, is to repeat what one time Mayor of Madrid, Ana Botella, famously said: "There is nothing quite like a relaxing cup of café con leche in Plaza Mayor".