Add ReactDOMClient to ServerIntegration(Hooks|NewContext) (#28135)

## Overview

Branched off https://github.com/facebook/react/pull/28130

### ~Failing~ Fixed by @eps1lon
Most of the tests pass, but there are 3 tests that have additional
warnings due to client render error retries.

For example, before we would log:

```
Warning: Do not call Hooks inside useEffect(...), useMemo(...), or other built-in Hooks.
Warning: Expected server HTML to contain a matching text node for "0" in <div>.
```

And now we log

```
Warning: Do not call Hooks inside useEffect(...), useMemo(...), or other built-in Hooks.
Warning: Expected server HTML to contain a matching text node for "0" in <div>.
Warning: Do not call Hooks inside useEffect(...), useMemo(...), or other built-in Hooks.
```

We can't just update the expected error count for these tests, because
the additional error only happens on the client. So I need some guidance
on how to fix these.

---------

Co-authored-by: Sebastian Silbermann <sebastian.silbermann@klarna.com>

DiffTrain build for commit https://github.com/facebook/react/commit/fa6674b5bcf52610e92d19a5105308e56091c386.

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react-test-renderer/cjs/ReactTestRenderer-dev.js

@@ -25622,7 +25622,7 @@ if (__DEV__) {
       return root;
     }
 
-    var ReactVersion = "18.3.0-canary-278199b3d-20240201";
+    var ReactVersion = "18.3.0-canary-fa6674b5b-20240201";
 
     // Might add PROFILE later.
 

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react-test-renderer/cjs/ReactTestRenderer-prod.js

@@ -9149,7 +9149,7 @@ var devToolsConfig$jscomp$inline_1031 = {
     throw Error("TestRenderer does not support findFiberByHostInstance()");
   },
   bundleType: 0,
-  version: "18.3.0-canary-278199b3d-20240201",
+  version: "18.3.0-canary-fa6674b5b-20240201",
   rendererPackageName: "react-test-renderer"
 };
 var internals$jscomp$inline_1207 = {
@@ -9180,7 +9180,7 @@ var internals$jscomp$inline_1207 = {
   scheduleRoot: null,
   setRefreshHandler: null,
   getCurrentFiber: null,
-  reconcilerVersion: "18.3.0-canary-278199b3d-20240201"
+  reconcilerVersion: "18.3.0-canary-fa6674b5b-20240201"
 };
 if ("undefined" !== typeof __REACT_DEVTOOLS_GLOBAL_HOOK__) {
   var hook$jscomp$inline_1208 = __REACT_DEVTOOLS_GLOBAL_HOOK__;

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react-test-renderer/cjs/ReactTestRenderer-profiling.js

@@ -9577,7 +9577,7 @@ var devToolsConfig$jscomp$inline_1073 = {
     throw Error("TestRenderer does not support findFiberByHostInstance()");
   },
   bundleType: 0,
-  version: "18.3.0-canary-278199b3d-20240201",
+  version: "18.3.0-canary-fa6674b5b-20240201",
   rendererPackageName: "react-test-renderer"
 };
 var internals$jscomp$inline_1248 = {
@@ -9608,7 +9608,7 @@ var internals$jscomp$inline_1248 = {
   scheduleRoot: null,
   setRefreshHandler: null,
   getCurrentFiber: null,
-  reconcilerVersion: "18.3.0-canary-278199b3d-20240201"
+  reconcilerVersion: "18.3.0-canary-fa6674b5b-20240201"
 };
 if ("undefined" !== typeof __REACT_DEVTOOLS_GLOBAL_HOOK__) {
   var hook$jscomp$inline_1249 = __REACT_DEVTOOLS_GLOBAL_HOOK__;

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react/cjs/React-dev.js

@@ -24,7 +24,7 @@ if (__DEV__) {
     ) {
       __REACT_DEVTOOLS_GLOBAL_HOOK__.registerInternalModuleStart(new Error());
     }
-    var ReactVersion = "18.3.0-canary-278199b3d-20240201";
+    var ReactVersion = "18.3.0-canary-fa6674b5b-20240201";
 
     // ATTENTION
     // When adding new symbols to this file,

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react/cjs/React-prod.js

@@ -545,4 +545,4 @@ exports.useSyncExternalStore = function (
 exports.useTransition = function () {
   return ReactCurrentDispatcher.current.useTransition();
 };
-exports.version = "18.3.0-canary-278199b3d-20240201";
+exports.version = "18.3.0-canary-fa6674b5b-20240201";

compiled-rn/facebook-fbsource/xplat/js/RKJSModules/vendor/react/cjs/React-profiling.js

@@ -541,7 +541,7 @@ exports.useSyncExternalStore = function (
 exports.useTransition = function () {
   return ReactCurrentDispatcher.current.useTransition();
 };
-exports.version = "18.3.0-canary-278199b3d-20240201";
+exports.version = "18.3.0-canary-fa6674b5b-20240201";
 "undefined" !== typeof __REACT_DEVTOOLS_GLOBAL_HOOK__ &&
   "function" ===
     typeof __REACT_DEVTOOLS_GLOBAL_HOOK__.registerInternalModuleStop &&

compiled-rn/facebook-fbsource/xplat/js/react-native-github/Libraries/Renderer/REVISION

@@ -1 +1 @@
-278199b3deeee73f969a242c777c2d38406f8bfe
+fa6674b5bcf52610e92d19a5105308e56091c386