[FEATURE] Add utm_source=claude-code to all urls

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

Website owners don't know how much traffic they are getting from Anthropics' applications. As more people are using AI for answering questions there is less search traffic and use of the browser and more coming from links within AI applications such as Claude Code. Adding a utm_source param would help publishers and website owners to understand where their traffic is coming from. For Anthropic this would also help those websites understand the value that Anthropic products are providing.

### Proposed Solution

Add a utm_source param to all urls that claude code is providing to the user.

Some suggested params to send:
- utm_source=claude-code
- utm_source=claude-cowork
- utm_source=claude-mobile

### Alternative Solutions

utm_source is a better choice than the http referral information because referral is often blocked by browsers and it is not available when going from an application into a browser.

The other utm_* params may also be useful, but start to run into privacy issues: https://en.wikipedia.org/wiki/UTM_parameters

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. A user asking claude to help me research and buy a new car
2. A number of links are given to the user to look at the company webpages on cars
3. The user clicks those links and explores the webpages
4. The car company examines their analytics and can then attribute some portion of car sales to users going through Claude.

### Additional Context

I work on https://www.parse.ly/ and https://jetpack.com/stats/ which run on millions of websites.

No files, entry points, or tests are identified in the issue. Start by locating the paths that provide URLs to Claude Code users and determine whether URL handling is centralized or spread across features. Define coverage for all affected link-producing paths, including the intended product-specific source values, and verify the resulting URLs.

Written by the indexing model from the issue text.