Some preparations for Irish coffee demand that the whiskey and sugar be caramelized together by heating them in a heat-proof glass over a burner and then topped with hot coffee and thick liquid cream. It takes some experience to get this heating step right.

Baileys coffee is simply coffee with added Baileys liquour (cream optional), like many other [coffee and alcohol] combinations.

Of course there are lots of other sources where "Irish coffee" is simply whiskey + sugar + coffee + (even whipped) cream, so without knowing what standards your pub/restaurant follows, it's nearly impossible to say whether you could have ordered Baileys coffee.

Some preparations for Irish coffee demand that the whiskey and sugar be caramelized together by heating them in a heat-proof glass over a burner and then topped with hot coffee and thick liquid cream. It takes some experience to get this heating step right. (Youtube video here)

Baileys coffee is simply coffee with added Baileys liquour (cream optional), like many other [coffee and alcohol] combinations.

Of course there are lots of other sources where "Irish coffee" is simply whiskey + sugar + coffee + (even whipped) cream, probably the "original" preparation served to warm up travellers, so without knowing what standards your pub/restaurant follows, it's nearly impossible to say whether you could have ordered Baileys coffee.