Updating the docstring disable_env_checker
diff --git a/gym/envs/registration.py b/gym/envs/registration.py
index 8b7a2ac83b8..5b9c8a4fd30 100644
--- a/gym/envs/registration.py
+++ b/gym/envs/registration.py
@@ -126,7 +126,7 @@ class EnvSpec:
     * max_episode_steps: The max number of steps that the environment can take before truncation
     * order_enforce: If to enforce the order of `reset` before `step` and `render` functions
     * autoreset: If to automatically reset the environment on episode end
-    * disable_env_checker: If to disable the environment checker wrapper by default in `gym.make`
+    * disable_env_checker: If to disable the environment checker wrapper in `gym.make`, by default False (runs the environment checker)
     * kwargs: Additional keyword arguments passed to the environments through `gym.make`
     """
 
@@ -558,8 +558,9 @@ def make(
         max_episode_steps: Maximum length of an episode (TimeLimit wrapper).
         autoreset: Whether to automatically reset the environment after each episode (AutoResetWrapper).
         new_step_api: Whether to use old or new step API (StepAPICompatibility wrapper). Will be removed at v1.0
-        disable_env_checker: If to run the env checker, None will default to the environment `spec.disable_env_checker`
-            (that is by default True), otherwise will run according to the parameter (True = not run, False = run)
+        disable_env_checker: If to run the env checker, None will default to the environment specification `disable_env_checker`
+            (which is by default False, running the environment checker),
+            otherwise will run according to this parameter (`True` = not run, `False` = run)
         kwargs: Additional arguments to pass to the environment constructor.
 
     Returns:
diff --git a/gym/vector/__init__.py b/gym/vector/__init__.py
index 185082715c8..3dc4998fa8d 100644
--- a/gym/vector/__init__.py
+++ b/gym/vector/__init__.py
@@ -35,9 +35,10 @@ def make(
         num_envs: Number of copies of the environment.
         asynchronous: If `True`, wraps the environments in an :class:`AsyncVectorEnv` (which uses `multiprocessing`_ to run the environments in parallel). If ``False``, wraps the environments in a :class:`SyncVectorEnv`.
         wrappers: If not ``None``, then apply the wrappers to each internal environment during creation.
-        disable_env_checker: If to disable the env checker, if True it will only run on the first environment created.
+        disable_env_checker: If to run the env checker for the first environment only. None will default to the environment spec `disable_env_checker` parameter
+            (that is by default False), otherwise will run according to this argument (True = not run, False = run)
         new_step_api: If True, the vector environment's step method outputs two booleans `terminated`, `truncated` instead of one `done`.
-        **kwargs: Keywords arguments applied during gym.make
+        **kwargs: Keywords arguments applied during `gym.make`
 
     Returns:
         The vectorized environment.
diff --git a/gym/vector/vector_env.py b/gym/vector/vector_env.py
index ad2710e02ae..3ca4663d822 100644
--- a/gym/vector/vector_env.py
+++ b/gym/vector/vector_env.py
@@ -36,7 +36,7 @@ def __init__(
             num_envs: Number of environments in the vectorized environment.
             observation_space: Observation space of a single environment.
             action_space: Action space of a single environment.
-            new_step_api (bool): Whether the vector env's step method outputs two boolean arrays (new API) or one boolean array (old API)
+            new_step_api (bool): Whether the vector environment's step method outputs two boolean arrays (new API) or one boolean array (old API)
         """
         self.num_envs = num_envs
         self.is_vector_env = True
@@ -54,8 +54,7 @@ def __init__(
         self.new_step_api = new_step_api
         if not self.new_step_api:
             deprecation(
-                "Initializing vector env in old step API which returns one bool array instead of two. "
-                "It is recommended to set `new_step_api=True` to use new step API. This will be the default behaviour in future. "
+                "Initializing vector env in old step API which returns one bool array instead of two. It is recommended to set `new_step_api=True` to use new step API. This will be the default behaviour in future."
             )
 
     def reset_async(
@@ -147,7 +146,7 @@ def step(self, actions):
             actions: element of :attr:`action_space` Batch of actions.
 
         Returns:
-            Batch of (observations, rewards, terminateds, truncateds, infos) or (observations, rewards, dones, infos)
+            Batch of (observations, rewards, terminated, truncated, infos) or (observations, rewards, dones, infos)
         """
         self.step_async(actions)
         return self.step_wait()

Address the comment in https://github.com/openai/gym/commit/519dfd9117e98e4f52d38064d2b0f79974fb676d