[BUG] /model default resolves to Opus 4.7 but UI picker shows Default = Sonnet 4.6

Found 3 possible duplicate issues:

1. https://github.com/anthropics/claude-code/issues/35789
2. https://github.com/anthropics/claude-code/issues/43261
3. https://github.com/anthropics/claude-code/issues/45978

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/model default resolving to Opus 4.7 while the VS Code extension UI picker labels Default as Sonnet 4.6 is an agent identity display mismatch — the user selects what they believe is "Sonnet 4.6 (default)" but the agent actually runs as Opus 4.7. The model that executes differs from the model the user believes they selected.

This is the same class of silent model substitution documented in:

• #49006: Opus thinking effort reduced without disclosure
• #49076: /model persists effortLevel to settings.json as side effect
• #50363: Scheduled tasks always run Sonnet regardless of UI selection

In all cases, the declared model configuration and the actual runtime model diverge without notification.

The three possible root causes:

1. The UI label is stale. The VS Code extension UI still shows "Sonnet 4.6" as the label for Default, but the server has changed the actual default to Opus 4.7. The UI needs to be updated to reflect the current server-side default.

1. The /model default command and the UI picker are reading from different sources. The CLI command reads the current server default; the UI reads a cached/hardcoded label. They need to read from the same source.

1. The "default" is being overridden by a stored model preference. If a previous session set Opus 4.7 as the active model and the setting persisted (related to #49076), /model default may be returning the persisted model rather than the actual default.

The fix: The "Default" option in the model picker should display the actual current default model identifier, not a hardcoded label. When the user selects Default, the agent should start on whatever model the current default is, and the UI should confirm which model that is.

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