chore(input): align codex tooling docs

input/Docker/run-input-processor.sh

@@ -9,27 +9,20 @@ fi
 
 if docker compose version >/dev/null 2>&1; then
   COMPOSE_CMD=(docker compose)
-elif command -v docker-compose >/dev/null 2>&1; then
+elif command -dv docker-compose >/dev/null 2>&1; then
   COMPOSE_CMD=(docker-compose)
 else
   echo "Error: docker compose plugin or docker-compose binary is required." >&2
   exit 1
 fi
 
-CALLER_UID=$(id -u)
-CALLER_GID=$(id -g)
-
-DEFAULT_CODEX_CONFIG="${HOME}/.codex"
-CODEX_CONFIG_DIR=${CODEX_CONFIG_DIR:-${DEFAULT_CODEX_CONFIG}}
-
-mkdir -p "${CODEX_CONFIG_DIR}"
-
 SCRIPT_DIR=$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)
+HOST_CODEX_DIR=${HOST_CODEX_DIR:-${HOME}/.codex}
+
+mkdir -p "${HOST_CODEX_DIR}"
 
 (
   cd "${SCRIPT_DIR}"
-  LOCAL_UID="${CALLER_UID}" \
-  LOCAL_GID="${CALLER_GID}" \
-  CODEX_CONFIG_DIR="${CODEX_CONFIG_DIR}" \
+  HOST_CODEX_DIR="${HOST_CODEX_DIR}" \
     "${COMPOSE_CMD[@]}" "$@"
 )