Model-theoretic characterizations of large cardinals (Re)²visited

We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that Πₙ-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vop{e}nka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vop{e}nka's Principle and weak Vop{e}nka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic L(QWF), and show that the compactness number of the H\"artig quantifier logic L(I) can consistently be larger than the first supercompact cardinal. Finally, we show that the upward L\"owenheim-Skolem-Tarski number of second-order logic L² and the sort logic Ls,n are given by the first extendible and C⁽ⁿ⁾-extendible cardinal, respectively.