A type is not an expression. We wouldn't have two words designating the same concept. Even in dependently typed languages where types and expressions are intermingled, they are still distinct concepts.
Now, you can sort of talk about expressions in the "type language", but these are not expressions of the "value language". Even so, an unqualified statement like "a type is an expression" is simply incorrect because "expression" always refers to the value language, so that phrase conveys the completely wrong intention.
Finally, as for how to relate staging to concepts you might be more familiar with, I suggest the paper, Closing the Stage: From Staged Code to Typed Closures [1].
This is only true in dependently typed languages. And even then, soundness requires stratifying types into universes or something similar.
> In other words, types are higher order expressions.
This isn't the meaning of higher order as it applies to programming languages or the standard isomorphism to logic.