Anchor Add Context picker to its chat input button

[meganrogge]

The Add Context quick pick currently opens in the standard centered quick-pick location, which disconnects it spatially from the chat input action that opened it.

Anchor Add Context to its button across chat inputs. The new-session composer should prefer placement below the button, while existing chat inputs should prefer placement above it. Placement should fall back to the opposite side when the preferred side does not fit, and activating the same button again should close the picker.

Implemented by #338924.

[vs-code-engineering]

Sharing media and context from merged PR #338924 for verification:

Anchors Add Context to its invoking button across chat inputs:

• The new-session composer prefers placement below the button.
• Existing chat inputs prefer placement above the button.
• Anchored quick inputs use standard spacing from their invoking control and fall back to the opposite side when the preferred side does not fit.
• Activating the same Add Context button again closes its picker, including in the new-session composer.

Validation:

• Targeted ESLint — passed
• ./scripts/test.sh --run src/vs/platform/quickinput/test/browser/quickinput.test.ts --grep "positions an anchored picker below" — passed
• ./scripts/test.sh --run src/vs/sessions/contrib/chat/test/browser/newChatInput.test.ts --grep "context picker" — passed
• npm run typecheck-client — passed

Source PR: #338924