refactor: extract update_claude_comment logic to standalone testable function

- Create new updateClaudeComment function in operations/comments
- Add comprehensive unit tests following image-downloader pattern
- Update MCP server to use extracted function
- Refactor update-comment-link.ts and update-with-branch.ts to eliminate duplication
- All tests passing (10 new tests for update-claude-comment)

Co-authored-by: ashwin-ant <ashwin-ant@users.noreply.github.com>

src/github/operations/comments/update-claude-comment.ts

@@ -0,0 +1,70 @@
+import { Octokit } from "@octokit/rest";
+
+export type UpdateClaudeCommentParams = {
+  owner: string;
+  repo: string;
+  commentId: number;
+  body: string;
+  isPullRequestReviewComment: boolean;
+};
+
+export type UpdateClaudeCommentResult = {
+  id: number;
+  html_url: string;
+  updated_at: string;
+};
+
+/**
+ * Updates a Claude comment on GitHub (either an issue/PR comment or a PR review comment)
+ * 
+ * @param octokit - Authenticated Octokit instance
+ * @param params - Parameters for updating the comment
+ * @returns The updated comment details
+ * @throws Error if the update fails
+ */
+export async function updateClaudeComment(
+  octokit: Octokit,
+  params: UpdateClaudeCommentParams,
+): Promise<UpdateClaudeCommentResult> {
+  const { owner, repo, commentId, body, isPullRequestReviewComment } = params;
+
+  let response;
+
+  try {
+    if (isPullRequestReviewComment) {
+      // Try PR review comment API first
+      response = await octokit.rest.pulls.updateReviewComment({
+        owner,
+        repo,
+        comment_id: commentId,
+        body,
+      });
+    } else {
+      // Use issue comment API (works for both issues and PR general comments)
+      response = await octokit.rest.issues.updateComment({
+        owner,
+        repo,
+        comment_id: commentId,
+        body,
+      });
+    }
+  } catch (error: any) {
+    // If PR review comment update fails with 404, fall back to issue comment API
+    if (isPullRequestReviewComment && error.status === 404) {
+      response = await octokit.rest.issues.updateComment({
+        owner,
+        repo,
+        comment_id: commentId,
+        body,
+      });
+    } else {
+      throw error;
+    }
+  }
+
+  return {
+    id: response.data.id,
+    html_url: response.data.html_url,
+    updated_at: response.data.updated_at,
+  };
+}