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/kubectl/portforward.go

@@ -74,7 +74,7 @@ func pfPod(expectedClientData, chunks, chunkSize, chunkIntervalMillis string, bi
 					Image: imageutils.GetE2EImage(imageutils.Agnhost),
 					Args:  []string{"netexec"},
 					ReadinessProbe: &v1.Probe{
-						Handler: v1.Handler{
+						ProbeHandler: v1.ProbeHandler{
 							Exec: &v1.ExecAction{
 								Command: []string{
 									"sh", "-c", "netstat -na | grep LISTEN | grep -v 8080 | grep 80",