docs: simplify model config in BROWSERBASE snippets to use string format

[chromiebot]

When using Browserbase, apiKey is now optional for openai/google/anthropic providers. Replace model object { modelName, apiKey } with simple string format "provider/model" to reduce friction in docs examples.

Updated files:

• best-practices/caching.mdx (1 instance)
• best-practices/deployments.mdx (1 instance)
• best-practices/deterministic-agent.mdx (4 instances)

why

what changed

test plan

---

Summary by cubic

Simplified Browserbase examples to use a single "provider/model" string instead of a { modelName, apiKey } object. Since apiKey is now optional for OpenAI, Google, and Anthropic on Browserbase, the docs are easier to copy and use.

^{Written for commit f55ff55. Summary will update on new commits. Review in cubic}

[changeset-bot]

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Latest commit: f55ff55

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[github-actions]

This PR is from an external contributor and must be approved by a stagehand team member with write access before CI can run.
Approving the latest commit mirrors it into an internal PR owned by the approver.
If new commits are pushed later, the internal PR stays open but is marked stale until someone approves the latest external commit and refreshes it.

[github-actions]

This PR was approved by @miguelg719 and mirrored to #1961. All further discussion should happen on that PR.

[cubic-dev-ai]

No issues found across 3 files

Confidence score: 5/5

• Automated review surfaced no issues in the provided summaries.
• No files require special attention.

Architecture diagram

sequenceDiagram
    participant User as Developer Code
    participant SDK as Stagehand SDK
    participant BB as Browserbase Infrastructure
    participant LLM as LLM Provider (Google/OpenAI/Anthropic)

    Note over User,LLM: Runtime Model Initialization Flow

    User->>SDK: stagehand.agent({ model: "provider/model" })
    Note right of User: CHANGED: Model defined as simple string<br/>instead of { modelName, apiKey } object

    SDK->>BB: Initialize session with model identifier
    
    Note over BB: NEW: Browserbase resolves LLM API key<br/>internally for supported providers

    BB->>LLM: Forward request with internal credentials
    LLM-->>BB: LLM Response (Inference/Action)
    
    BB-->>SDK: Return execution results
    SDK-->>User: Agent response

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[github-actions]

The mirrored PR #1961 has been merged into main. This original external contributor PR will stay closed as completed.