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

test/e2e/framework/network/utils.go

@@ -558,7 +558,7 @@ func (config *NetworkingTestConfig) createNetShellPodSpec(podName, hostname stri
 		PeriodSeconds:       10,
 		SuccessThreshold:    1,
 		FailureThreshold:    3,
-		Handler: v1.Handler{
+		ProbeHandler: v1.ProbeHandler{
 			HTTPGet: &v1.HTTPGetAction{
 				Path: "/healthz",
 				Port: intstr.IntOrString{IntVal: EndpointHTTPPort},