I am getting two notions confused with one another, and would be grateful for a clarification.

I take as an example the large cardinal axiom that an uncountable measurable cardinal exists. Measurable cardinals are Pi(2,1)- indescribable (superscript, subscript), which would mean that the axiom needs to quantify over classes. This would seem to put it into the realm of second-order logic so that, for example, the Compactness and Löwenheim-Skolem Theorems would not apply. But I feel that this is not correct. Please point out my mistake. Thank you.