diff --git a/README.md b/README.md
index 8da3982..57d4dc7 100644
--- a/README.md
+++ b/README.md
@@ -23,10 +23,11 @@ The library supports the following methods of retrieving tokens in the listed pr
 1. Reading custom service account credentials from the path pointed to by the
    `GOOGLE_APPLICATION_CREDENTIALS` environment variable. Alternatively, custom service
    account credentials can be read from a JSON file or string.
-2. Retrieving a token from the `gcloud` CLI tool, if it is available on the `PATH`.
+2. Look for credentials in `.config/gcloud/application_default_credentials.json`;
+   if found, use these credentials to request refresh tokens. This file can be created
+   by invoking `gcloud auth application-default login`.
 3. Use the default service account by retrieving a token from the metadata server.
-4. Look for credentials in `.config/gcloud/application_default_credentials.json`;
-   if found, use these credentials to request refresh tokens.
+4. Retrieving a token from the `gcloud` CLI tool, if it is available on the `PATH`.
 
 For more detailed information and examples, see the [docs][docs-url].
 
diff --git a/src/authentication_manager.rs b/src/authentication_manager.rs
index 7150c49..cd364ab 100644
--- a/src/authentication_manager.rs
+++ b/src/authentication_manager.rs
@@ -34,12 +34,12 @@ impl AuthenticationManager {
     ///
     /// 1. Check if the `GOOGLE_APPLICATION_CREDENTIALS` environment variable if set;
     ///    if so, use a custom service account as the token source.
-    /// 2. Check if the `gcloud` tool is available on the `PATH`; if so, use the
-    ///    `gcloud auth print-access-token` command as the token source.
+    /// 2. Look for credentials in `.config/gcloud/application_default_credentials.json`;
+    ///    if found, use these credentials to request refresh tokens.
     /// 3. Send a HTTP request to the internal metadata server to retrieve a token;
     ///    if it succeeds, use the default service account as the token source.
-    /// 4. Look for credentials in `.config/gcloud/application_default_credentials.json`;
-    ///    if found, use these credentials to request refresh tokens.
+    /// 4. Check if the `gcloud` tool is available on the `PATH`; if so, use the
+    ///    `gcloud auth print-access-token` command as the token source.
     #[tracing::instrument]
     pub async fn new() -> Result<Self, Error> {
         tracing::debug!("Initializing gcp_auth");
@@ -48,9 +48,9 @@ impl AuthenticationManager {
         }
 
         let client = types::client();
-        let gcloud_error = match GCloudAuthorizedUser::new().await {
+        let default_user_error = match DefaultAuthorizedUser::new(&client).await {
             Ok(service_account) => {
-                tracing::debug!("Using GCloudAuthorizedUser");
+                tracing::debug!("Using DefaultAuthorizedUser");
                 return Ok(Self::build(client, service_account));
             }
             Err(e) => e,
@@ -64,9 +64,9 @@ impl AuthenticationManager {
             Err(e) => e,
         };
 
-        let default_user_error = match DefaultAuthorizedUser::new(&client).await {
+        let gcloud_error = match GCloudAuthorizedUser::new().await {
             Ok(service_account) => {
-                tracing::debug!("Using DefaultAuthorizedUser");
+                tracing::debug!("Using GCloudAuthorizedUser");
                 return Ok(Self::build(client, service_account));
             }
             Err(e) => e,
diff --git a/src/lib.rs b/src/lib.rs
index 400a491..08c29c5 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -10,10 +10,11 @@
 //! 1. Reading custom service account credentials from the path pointed to by the
 //!    `GOOGLE_APPLICATION_CREDENTIALS` environment variable. Alternatively, custom service
 //!    account credentials can be read from a JSON file or string.
-//! 2. Retrieving a token from the `gcloud` CLI tool, if it is available on the `PATH`.
+//! 2. Look for credentials in `.config/gcloud/application_default_credentials.json`;
+//!    if found, use these credentials to request refresh tokens. This file can be created
+//!    by invoking `gcloud auth application-default login`.
 //! 3. Use the default service account by retrieving a token from the metadata server.
-//! 4. Look for credentials in `.config/gcloud/application_default_credentials.json`;
-//!    if found, use these credentials to request refresh tokens.
+//! 4. Retrieving a token from the `gcloud` CLI tool, if it is available on the `PATH`.
 //!
 //! For more details, see [`AuthenticationManager::new()`].
 //!