[Embeddable] EUI Visual Refresh Integration (elastic#204452)

## Summary

Related to elastic#203132.
Part of [elastic#204596](elastic#204596).

This replaces all references to euiThemeVars in favor of the useEuiTheme
hook.

### Checklist

Check the PR satisfies following conditions. 

Reviewers should verify this PR satisfies this list as well.

- [ ] Any text added follows [EUI's writing
guidelines](https://elastic.github.io/eui/#/guidelines/writing), uses
sentence case text and includes [i18n
support](https://github.com/elastic/kibana/blob/main/packages/kbn-i18n/README.md)
- [ ]
[Documentation](https://www.elastic.co/guide/en/kibana/master/development-documentation.html)
was added for features that require explanation or tutorials
- [ ] [Unit or functional
tests](https://www.elastic.co/guide/en/kibana/master/development-tests.html)
were updated or added to match the most common scenarios
- [ ] If a plugin configuration key changed, check if it needs to be
allowlisted in the cloud and added to the [docker
list](https://github.com/elastic/kibana/blob/main/src/dev/build/tasks/os_packages/docker_generator/resources/base/bin/kibana-docker)
- [ ] This was checked for breaking HTTP API changes, and any breaking
changes have been approved by the breaking-change committee. The
`release_note:breaking` label should be applied in these situations.
- [ ] [Flaky Test
Runner](https://ci-stats.kibana.dev/trigger_flaky_test_runner/1) was
used on any tests changed
- [ ] The PR description includes the appropriate Release Notes section,
and the correct `release_note:*` label is applied per the
[guidelines](https://www.elastic.co/guide/en/kibana/master/contributing.html#kibana-release-notes-process)

### Identify risks

Does this PR introduce any risks? For example, consider risks like hard
to test bugs, performance regression, potential of data loss.

Describe the risk, its severity, and mitigation for each identified
risk. Invite stakeholders and evaluate how to proceed before merging.

- [ ] [See some risk
examples](https://github.com/elastic/kibana/blob/main/RISK_MATRIX.mdx)
- [ ] ...

---------

Co-authored-by: kibanamachine <[email protected]>
Co-authored-by: Elastic Machine <[email protected]>

examples/embeddable_examples/public/react_embeddables/eui_markdown/eui_markdown_react_embeddable.tsx

@@ -7,7 +7,7 @@
  * License v3.0 only", or the "Server Side Public License, v 1".
  */
 
-import { EuiMarkdownEditor, EuiMarkdownFormat } from '@elastic/eui';
+import { EuiMarkdownEditor, EuiMarkdownFormat, useEuiTheme } from '@elastic/eui';
 import { css } from '@emotion/react';
 import { ReactEmbeddableFactory } from '@kbn/embeddable-plugin/public';
 import { i18n } from '@kbn/i18n';
@@ -16,7 +16,6 @@ import {
   useInheritedViewMode,
   useStateFromPublishingSubject,
 } from '@kbn/presentation-publishing';
-import { euiThemeVars } from '@kbn/ui-theme';
 import React from 'react';
 import { BehaviorSubject } from 'rxjs';
 import { EUI_MARKDOWN_ID } from './constants';
@@ -80,6 +79,7 @@ export const markdownEmbeddableFactory: ReactEmbeddableFactory<
         // get state for rendering
         const content = useStateFromPublishingSubject(content$);
         const viewMode = useInheritedViewMode(api) ?? 'view';
+        const { euiTheme } = useEuiTheme();
 
         return viewMode === 'edit' ? (
           <EuiMarkdownEditor
@@ -96,7 +96,7 @@ export const markdownEmbeddableFactory: ReactEmbeddableFactory<
         ) : (
           <EuiMarkdownFormat
             css={css`
-              padding: ${euiThemeVars.euiSizeM};
+              padding: ${euiTheme.size.m};
             `}
           >
             {content ?? ''}

examples/embeddable_examples/public/react_embeddables/field_list/field_list_react_embeddable.tsx

@@ -7,7 +7,7 @@
  * License v3.0 only", or the "Server Side Public License, v 1".
  */
 
-import { EuiFlexGroup, EuiFlexItem } from '@elastic/eui';
+import { EuiFlexGroup, EuiFlexItem, useEuiTheme } from '@elastic/eui';
 import { css } from '@emotion/react';
 import type { Reference } from '@kbn/content-management-utils';
 import { CoreStart } from '@kbn/core-lifecycle-browser';
@@ -17,7 +17,6 @@ import { ReactEmbeddableFactory } from '@kbn/embeddable-plugin/public';
 import { i18n } from '@kbn/i18n';
 import { initializeTitles, useBatchedPublishingSubjects } from '@kbn/presentation-publishing';
 import { LazyDataViewPicker, withSuspense } from '@kbn/presentation-util-plugin/public';
-import { euiThemeVars } from '@kbn/ui-theme';
 import {
   UnifiedFieldListSidebarContainer,
   type UnifiedFieldListSidebarContainerProps,
@@ -150,6 +149,7 @@ export const getFieldListFactory = (
             dataViews$,
             selectedFieldNames$
           );
+          const { euiTheme } = useEuiTheme();
 
           const selectedDataView = renderDataViews?.[0];
 
@@ -165,7 +165,7 @@ export const getFieldListFactory = (
               <EuiFlexItem
                 grow={false}
                 css={css`
-                  padding: ${euiThemeVars.euiSizeS};
+                  padding: ${euiTheme.size.s};
                 `}
               >
                 <DataViewPicker

examples/embeddable_examples/public/react_embeddables/saved_book/saved_book_react_embeddable.tsx

@@ -7,7 +7,15 @@
  * License v3.0 only", or the "Server Side Public License, v 1".
  */
 
-import { EuiBadge, EuiCallOut, EuiFlexGroup, EuiFlexItem, EuiText, EuiTitle } from '@elastic/eui';
+import {
+  EuiBadge,
+  EuiCallOut,
+  EuiFlexGroup,
+  EuiFlexItem,
+  EuiText,
+  EuiTitle,
+  useEuiTheme,
+} from '@elastic/eui';
 import { css } from '@emotion/react';
 import { CoreStart } from '@kbn/core-lifecycle-browser';
 import { ReactEmbeddableFactory } from '@kbn/embeddable-plugin/public';
@@ -18,7 +26,6 @@ import {
   SerializedTitles,
   useBatchedPublishingSubjects,
 } from '@kbn/presentation-publishing';
-import { euiThemeVars } from '@kbn/ui-theme';
 import React from 'react';
 import { BehaviorSubject } from 'rxjs';
 import { PresentationContainer } from '@kbn/presentation-containers';
@@ -157,6 +164,7 @@ export const getSavedBookEmbeddableFactory = (core: CoreStart) => {
               bookAttributesManager.bookTitle,
               bookAttributesManager.bookSynopsis
             );
+          const { euiTheme } = useEuiTheme();
 
           return (
             <div
@@ -182,7 +190,7 @@ export const getSavedBookEmbeddableFactory = (core: CoreStart) => {
               )}
               <div
                 css={css`
-                  padding: ${euiThemeVars.euiSizeM};
+                  padding: ${euiTheme.size.m};
                 `}
               >
                 <EuiFlexGroup

examples/embeddable_examples/public/react_embeddables/search/search_react_embeddable.tsx

@@ -7,7 +7,7 @@
  * License v3.0 only", or the "Server Side Public License, v 1".
  */
 
-import { EuiBadge, EuiStat } from '@elastic/eui';
+import { EuiBadge, EuiStat, useEuiTheme } from '@elastic/eui';
 import { css } from '@emotion/react';
 import { DataView } from '@kbn/data-views-plugin/common';
 import { ReactEmbeddableFactory } from '@kbn/embeddable-plugin/public';
@@ -17,7 +17,6 @@ import {
   initializeTimeRange,
   useBatchedPublishingSubjects,
 } from '@kbn/presentation-publishing';
-import { euiThemeVars } from '@kbn/ui-theme';
 import React, { useEffect } from 'react';
 import { BehaviorSubject, switchMap, tap } from 'rxjs';
 import { SEARCH_EMBEDDABLE_ID } from './constants';
@@ -124,6 +123,7 @@ export const getSearchEmbeddableFactory = (services: Services) => {
         api,
         Component: () => {
           const [count, error] = useBatchedPublishingSubjects(count$, blockingError$);
+          const { euiTheme } = useEuiTheme();
 
           useEffect(() => {
             return () => {
@@ -138,7 +138,7 @@ export const getSearchEmbeddableFactory = (services: Services) => {
             <div
               css={css`
                 width: 100%;
-                padding: ${euiThemeVars.euiSizeM};
+                padding: ${euiTheme.size.m};
               `}
             >
               <EuiStat

examples/embeddable_examples/tsconfig.json

@@ -18,7 +18,6 @@
     "@kbn/dashboard-plugin",
     "@kbn/embeddable-plugin",
     "@kbn/presentation-publishing",
-    "@kbn/ui-theme",
     "@kbn/i18n",
     "@kbn/es-query",
     "@kbn/data-views-plugin",