[data.search.SearchSource] Unify FetchHandler types & add onResponse/callMsearch.

[lukeelmers]

Part of #65069

Summary

This PR cleans up a few remaining items so that SearchSource can be moved into common and shared between client & server:

• Removes usage of handleResponse and replaces with a new onResponse callback that's provided to SearchSource
  • handleResponse is used in the client to display errors in the UI; on the server it will just pass along the response unchanged
• Removes usage of core.http which has different client & server APIs, and replaces with callMsearch
  • callMsearch is a simple helper for calling the /internal/_msearch endpoint, and abstracts away the usage of the http service
  • On the server we will provide a different implementation of this same function
  • This is only used in the legacy default search strategy
• Renames the config in FetchHandlers to getConfig for consistency with other areas of the data plugin
• Extends the SearchSourceDependencies from FetchHandlers to remove duplicated types
• Groups callMsearch and loadingCount$ under legacy in FetchHandlers, to make it clear what configuration can be removed from SearchSource whenever the legacy default search strategy is removed.

Testing

This is a refactoring; no functional changes were introduced. Most of the updates here affect the behavior of Discover / Visualizations / Dashboards when the courier:batchSearches advanced setting is enabled.

[elasticmachine]

Pinging @elastic/kibana-app-arch (Team:AppArch)

[ppisljar]

code LGTM

[kibanamachine]

💚 Build Succeeded

• continuous-integration/kibana-ci/pull-request
• Commit: 08ffb22

Build metrics

@kbn/optimizer bundle module count

id	value	diff	baseline
data	543	+1	542

page load bundle size

id	value	diff	baseline
data	1.4MB	+1.2KB	1.4MB

To update your PR or re-run it, just comment with:
@elasticmachine merge upstream