Why does `defer recover()` not catch panics?

Quoting from the documentation of the built-in function recover():

If recover is called outside the deferred function it will not stop a panicking sequence.

In your second case recover() itself is the deferred function, and obviously recover() does not call itself. So this will not stop the panicking sequence.

If recover() would call recover() in itself, it would stop the panicking sequence (but why would it do that?).

Another Interesting Example:

The following code also doesn't panic (try it on the Go Playground):

package main

func main() {
    var recover = func() { recover() }
    defer recover()
    panic("panic")
}

What happens here is we create a recover variable of function type which has a value of an anonymous function calling the built-in recover() function. And we specify calling the value of the recover variable to be the deferred function, so calling the builtin recover() from that stops the panicing sequence.

The Handling panic section mentions that

Two built-in functions, panic and recover, assist in reporting and handling run-time panics

The recover function allows a program to manage behavior of a panicking goroutine.

Suppose a function G defers a function D that calls recover and a panic occurs in a function on the same goroutine in which G is executing.

When the running of deferred functions reaches D, the return value of D's call to recover will be the value passed to the call of panic.
If D returns normally, without starting a new panic, the panicking sequence stops.

That illustrates that recover is meant to be called in a deferred function, not directly.
When it panic, the "deferred function" cannot be the built-in recover() one, but one specified in a defer statement.

DeferStmt = "defer" Expression .

The expression must be a function or method call; it cannot be parenthesized.
Calls of built-in functions are restricted as for expression statements.

With the exception of specific built-in functions, function and method calls and receive operations can appear in statement context.