Hola, Stuart:

Por lo general, lo puedes dintinguir por el contexto en el que se usa la frase. Sin embargo, si el contexto no fuera claro y quisieras hacerlo más obvio podrías decir: 

- 'Hoy rompí una copa de las del vino' ('Hoy rompí una copa de las que usamos para el vino').

En una situación común, en la que la copa o vaso no tuvieran líquido dentro, diríamos:

'Rompí una copa' o 'Rompí un vaso'

No solemos especificar ya que las copas suelen ser usadas para vino y los vasos para agua, refrescos... aunque esto ya depende de cada persona y los usos que le dé a cada cosa.

Espero que te sirva de ayuda.

Thank you for taking the time to comment. Much appreciated.

The name of that kind of containers is "copa". We use the "copas" for the wine and we use "vasos (glasses)" for other kind of liquids as the orange juice.

If you say "Rompí una copa de vino" the context helps to understand if the cup was with wine or not inside. In other hand, you have to specify if the cup was with wine inside or not.

For example: If you were in the bar, and you tell your friends you were drinking a wine of glass "una copa de vino" and it fell down and it broke itself. Everyone is going to understand that the glass was full of wine.

But if you say: I was washing my plates and glasses in the kitchen and the glass fell down from my hands. It's almost sure they are going to understand the glass was emptied.

But you can be more specific as Brenda said you before :)

Five moths! you have a great Spanish. Congratulations