For "spiced" we also have the word "seasoned." It has a handful of meanings, but if you're using it to describe a flavor it should be clear that's the meaning you're using.
I've also heard foods described as having "bite" to indicate that they have a strong, spicy flavor, but not necessarily a "hot" one.
The closest I think in English is how we have modified the word "spice" to be, in context, "spiced" vs "spicy".
The former implies the "herbs and spices" and the latter the "hot pepper" types. But it's not exactly consistent.
We of course have the modifier "hot" which works. And we stole piquant from the french, but that is a little different I think.