It's not quite as simple as that. Even the same sentence can mean either:
Since you left me, I've been lonely. ->
a. From the time you left me, I've been lonely.
b. Because you left me, I've been lonely.
The normal usage is a. But that doesn't make b. wrong. Well, it doesn't make it wrong in principle but I don't think anyone would understand this to mean b, would they?