chore: standardize Claude Code CLI naming (#915)

Follow-up to #915.

docs/gateway/cli-backends.md

@@ -2,7 +2,7 @@
 summary: "CLI backends: text-only fallback via local AI CLIs"
 read_when:
   - You want a reliable fallback when API providers fail
-  - You are running Claude CLI or other local AI CLIs and want to reuse them
+  - You are running Claude Code CLI or other local AI CLIs and want to reuse them
   - You need a text-only, tool-free path that still supports sessions and images
 ---
 # CLI backends (fallback runtime)
@@ -20,7 +20,7 @@ want “always works” text responses without relying on external APIs.
 
 ## Beginner-friendly quick start
 
-You can use Claude CLI **without any config** (Clawdbot ships a built-in default):
+You can use Claude Code CLI **without any config** (Clawdbot ships a built-in default):
 
 ```bash
 clawdbot agent --message "hi" --model claude-cli/opus-4.5
@@ -162,7 +162,7 @@ imageMode: "repeat"
 Clawdbot will write base64 images to temp files. If `imageArg` is set, those
 paths are passed as CLI args. If `imageArg` is missing, Clawdbot appends the
 file paths to the prompt (path injection), which is enough for CLIs that auto-
-load local files from plain paths (Claude CLI behavior).
+load local files from plain paths (Claude Code CLI behavior).
 
 ## Inputs / outputs