When should you actually reach for the Adapter pattern instead of just modifying one of the two mismatched classes directly?

Adapter earns its place specifically when you don't control one side of the mismatch -- most commonly a third-party library or a legacy system you're not allowed to change. In those cases there's no option to rename a method or reorder its parameters, so a small translating class in between is the only way to make the incompatible class usable through the interface your application already depends on. If you own both classes involved in the mismatch, reaching for an adapter usually just adds an unnecessary layer of indirection; it's simpler to fix the signature mismatch directly. Adapter is a workaround for something outside your control, not a general-purpose way to connect any two interfaces.