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/storage/persistent_volumes.go

@@ -341,7 +341,7 @@ var _ = utils.SIGDescribe("PersistentVolumes", func() {
 				writeCmd += "&& sleep 10000"
 
 				probe := &v1.Probe{
-					Handler: v1.Handler{
+					ProbeHandler: v1.ProbeHandler{
 						Exec: &v1.ExecAction{
 							// Check that the last file got created
 							Command: []string{"test", "-f", getVolumeFile(numVols - 1)},