if it's a conditional clause it would be 条件節 じょうけんせつ
a conditional statement would be 条件文
Conditionals should really be taught BY THEIR FUNCTION class. Zero, 1st, 2nd and 3rd conditionals.
Perhaps an update to that linked page would be handy (it lists there being only 3 types when there are 4)

条件付 じょうけんつき is something different and unrelated as a grammar point. It is more like "strings attached" or "conditionally" as in "I agree conditionally"