It is often served in a special liqueur coffee glass with the top of the rich cream.
This coffee is very popular in Galicia, Spain where they considered this coffee as their traditional drink and there it is known as guajillo, but instead of Baileys, they put strong alcohol such as aguardiente or rum.
There are a lot of types of liqueur coffees and their name varies from country to country. In some cases, the same name may be used for the different combinations or vice versa.
For example you have Whiskey coffee or Gaelic coffee (with whiskey), Irish cream coffee (with Baileys), French coffee (with Grand Marnier), Brandy coffee, German coffee (with schnapps), Italian classic (with amaretto), English coffee (with gin) Shin Shin coffee (with rum), American coffee (with bourbon), The Real Foul One (with absinthe), Russian coffee (with vodka), Skye coffee (with Drambuie) and many many others.
So we may conclude that there is no unique recipe for creating the liqueur coffee, but ingredients adapt according to the things we have in the kitchen at a specific moment. Go ahead and create your own liqueur coffee.