Berry widget demo

[chvik]

Purpose of this pull request

This change integrates the Berry widget based on #4748 with the following updates:

• Adds the Berry widget inline on the homepage and as a popup elsewhere.
• Synchronizes the Berry widget's color mode with the site's color mode.
• Increases the height of the inline widget.
• Removes the resize button from the inline widget.

Select the type of change

• Minor Changes - Typos, formatting, slight revisions
• Update Content - Revisions, updating sections
• New Content - New features, sections, pages, tutorials
• Site and Tools - .clabot, version updates, maintenance, dependencies, new packages for the site (Docusaurus, Gatsby, React, etc.)

Ticket (if applicable)

[cla-bot]

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