De-share the Handler struct in core API (#105979)

* De-share the Handler struct in core API

An upcoming PR adds a handler that only applies on one of these paths.
Having fields that don't work seems bad.

This never should have been shared.  Lifecycle hooks are like a "write"
while probes are more like a "read". HTTPGet and TCPSocket don't really
make sense as lifecycle hooks (but I can't take that back). When we add
gRPC, it is EXPLICITLY a health check (defined by gRPC) not an arbitrary
RPC - so a probe makes sense but a hook does not.

In the future I can also see adding lifecycle hooks that don't make
sense as probes.  E.g. 'sleep' is a common lifecycle request. The only
option is `exec`, which requires having a sleep binary in your image.

* Run update scripts

pkg/kubelet/prober/common_test.go

@@ -75,7 +75,7 @@ func getTestPod() *v1.Pod {
 
 func setTestProbe(pod *v1.Pod, probeType probeType, probeSpec v1.Probe) {
 	// All tests rely on the fake exec prober.
-	probeSpec.Handler = v1.Handler{
+	probeSpec.ProbeHandler = v1.ProbeHandler{
 		Exec: &v1.ExecAction{},
 	}