I'm working through Kunen's Set Theory. In the chapter of cardinal arithmetic we define the cofinality of a limit ordinal as $$\text{cf}(\gamma) = \min \{\text{type}(X): X \subseteq\gamma \wedge \sup(X) = \gamma\}.$$
The way I interpret this definition is the cofinality is the smallest unbounded subset of $\gamma$. Or as the smallest subset that "approaches" $\gamma$. My question is why do we define it as the smallest order type of a subset? It seems like it would be more relevant, at least to this chapter, to define it as the smallest cardinality of a subset, i.e. $$\text{cf}(\gamma) = \min \{|X|: X \subseteq\gamma \wedge \sup(X) = \gamma\}.$$ This definition preserves the classes of regular cardinals and singular cardinals. Are these two definitions equivalent? Is there a reason why we prefer one over the other? Does defining cofinality with the order type serve a broader utility than just to show which cardinals are regular and which are singular?