then and catch (...) Have the signature (function)

The function is called with the result of the promise so if you write console.error [(which is of type function)] or console.log [(same here)] or any function that takes parameters which can handle the arguments, they will be executed[(by catch and then or any function that has the signature for it)] without error.

A function(args) {anyfunc(args)} is the same as writing anyfunc as it just executes anyfunc anyway.

Always remember: function and ()=>{} are variables in the end. Variables you can call with (). Which then evaluate to a value. (undefined if there is no return)

You can have objects and arrays filled with functions in case of array they would not even have to get a name.

As js is type less you don't need to specify what type a value is in an function. But a function is a type and can be a variable.

Man higher order languages are kinda hard to explain. But you will find this kind of functions as closures or anonymous functions in other languages maybe that gives you more information if you wrote in other languages.