Widgets: Search: Unify Jetpack and WP.com Code

[mdawaffe]

Changes proposed in this Pull Request:

• Extract default name into constructor.
• Extract search activation into methods. (Making it easy for WordPress.com
  to implement its own methods.)
• Fix some typos.

Testing instructions:

1. Create a site with Jetpack Professional plan
2. Go to https://wordpress.com/settings/traffic/{your site}
3. Make sure Jetpack Search (bottom of page) is disabled
4. Go to wp-admin/widgets.php
5. Add the "Search (Jetpack)" widget.
6. Go back to https://wordpress.com/settings/traffic/{your site} (reload the page)
7. (Wait a second or two)
8. See that Jetpack Search is enabled.

Proposed changelog entry for your changes:

Minor Changes to Jetpack Search Widget

[jetpackbot]

	Warnings
⚠️	The PR is missing at least one [Status] label. Suggestions: [Status] In Progress, [Status] Needs Review

This is automated check which relies on PULL_REQUEST_TEMPLATE.We encourage you to follow that template as it helps Jetpack maintainers do their job. If you think 'Testing instructions' or 'Proposed changelog entry' are not needed for your PR - please explain why you think so. Thanks for cooperation 🤖

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[gibrown]

WFM and LGTM

[jeherve]

Looking good, working well. Merging.