feat: add ClientRequestContext type alias for client-side handlers

[Kludex]

Summary

Following the refactoring in #1987 that split RequestContext between server and client, this PR introduces a dedicated ClientRequestContext type alias to improve the developer experience for client-side callback handlers.

Key changes:

• Added ClientRequestContext type alias in mcp.client.context for cleaner client-side handler signatures
• Made the internal RequestContext private by moving it to mcp.shared._context
• Exported ClientRequestContext from mcp.client for convenient access
• Updated all examples, documentation, and tests to use the new patterns

Before:

from mcp.shared.context import RequestContext
from mcp.client.session import ClientSession

async def handle_sampling(
    context: RequestContext[ClientSession],
    params: CreateMessageRequestParams
) -> CreateMessageResult:
    ...

After:

from mcp.client.context import ClientRequestContext

async def handle_sampling(
    context: ClientRequestContext,
    params: CreateMessageRequestParams
) -> CreateMessageResult:
    ...

Test plan

• All existing tests pass
• Ruff linting passes
• Pyright type checking passes
• Pre-commit hooks pass

[claude]

Code review

No issues found. Checked for bugs and CLAUDE.md compliance.