Update web to search interstitial transition

[joshliebe]

Task/Issue URL: https://app.asana.com/1/137249556945/project/488551667048375/task/1210498773096186?focus=true

Description

• Updates the transition from the browser tab to search interstitial

Steps to test this PR

• Enable Duck.ai PoC
• Tap the omnibar on browser tab
• Observe the transition to the search interstitial
• Verify that entered text is sent to the interstitial
• Go back
• Observe the transition to the browser tab
• Verify that entered text is sent back to the browser tab

[joshliebe]

• Update web to search interstitial transition #6204 👈 (View in Graphite)
• develop

This stack of pull requests is managed by Graphite. Learn more about stacking.

[Copilot]

Pull Request Overview

This PR updates the transition behavior from the browser tab to the search interstitial, ensuring that the entered query is passed correctly between activities and that shared element transitions are used for a smoother UI experience.

• Added a transition name ("omnibar_transition") to the omnibar layouts.
• Modified the SearchInterstitialActivity to retrieve and return query data via activity results.
• Updated the BrowserTabFragment to launch the interstitial with a shared element transition.

Reviewed Changes

Copilot reviewed 7 out of 7 changed files in this pull request and generated no comments.

Show a summary per file

File	Description
duckchat-impl/src/main/res/layout/view_duck_chat_omnibar.xml	Added transitionName attribute for shared element transitions.
duckchat-impl/src/main/java/com/duckduckgo/duckchat/impl/ui/SearchInterstitialActivity.kt	Updated activity parameters and result handling to support interstitial navigation with shared transitions.
duckchat-impl/src/main/java/com/duckduckgo/duckchat/impl/ui/DuckChatOmnibarLayout.kt	Removed text clearing from duckChatInput to preserve input state.
app/src/main/res/layout/view_fade_omnibar.xml	Added transitionName attribute for shared element transitions.
app/src/main/java/com/duckduckgo/app/browser/omnibar/experiments/FadeOmnibarLayout.kt	Changed access modifier for omniBarClickCatcher, removing the click listener to delegate handling.
app/src/main/java/com/duckduckgo/app/browser/omnibar/Omnibar.kt	Exposed omnibar container and click catcher for transition handling.
app/src/main/java/com/duckduckgo/app/browser/BrowserTabFragment.kt	Added activity result launcher and click catcher configuration to support the new interstitial transition with shared elements.

[aitorvs]

LGTM
Left a nit, maybe we want to add a TODO in kdoc to signal we may want to come back to the nit comment.
Works as expected