[Claimed #1959] docs: simplify model config in BROWSERBASE snippets to use string format

[github-actions]

Mirrored from external contributor PR #1959 after approval by @miguelg719.

Original author: @chromiebot
Original PR: #1959
Approved source head SHA: f55ff5558f2454e9f266b7f29ef0b511bd891b9e

@chromiebot, please continue any follow-up discussion on this mirrored PR. When the external PR gets new commits, this same internal PR will be marked stale until the latest external commit is approved and refreshed here.

Original description

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

Switch Browserbase docs snippets to use a single provider/model string for model config, replacing the { modelName, apiKey } object and omitting apiKey where it’s optional for OpenAI, Google, and Anthropic.
Updated examples across Basics (act, agent) and Best Practices (caching, deployments, deterministic-agent), including refreshed Google model IDs like google/gemini-3-flash-preview and google/gemini-2.5-flash.

^{Written for commit 7dc7863. Summary will update on new commits. Review in cubic}

[github-actions]

This mirrored PR has been merged into main. The original external PR #1959 is now completed.

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Latest commit: 7dc7863

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Project	Status	Preview	Updated (UTC)
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[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 App as User Application
    participant SDK as Stagehand SDK
    participant BB as Browserbase Platform
    participant LLM as LLM Provider (Google/OpenAI/Anthropic)

    Note over App,LLM: Model Configuration & Execution Flow

    App->>SDK: CHANGED: stagehand.agent({ model: "provider/model" })
    Note right of App: No longer requires explicit apiKey object

    SDK->>SDK: Parse provider and modelName from string

    App->>SDK: agent.act({ task: "..." })
    
    SDK->>BB: Initialize Session / Execute Action
    Note right of SDK: Passes model string to Browserbase

    BB->>BB: Resolve LLM Credentials
    
    alt NEW: API Key not provided by client
        BB->>BB: Use Browserbase Managed API Keys
    else API Key provided in config
        BB->>BB: Use Client-provided API Key
    end

    BB->>LLM: Request Inference (model, prompt)
    LLM-->>BB: Response (actions/text)
    BB-->>SDK: Action result
    SDK-->>App: Task completed

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