gh-70795: Rework RLock documentation (#103853)

Attempted to simultaneously reduce verbosity, while more descriptively
describing behavior.

Fix links (RLock acquire/release previously linking to Lock
acquire/release, seems like bad copy pasta).

Add a seealso for with-locks.

Switch section to use bullet points.

---------

Co-authored-by: Alex Waygood <Alex.Waygood@Gmail.com>
Co-authored-by: C.A.M. Gerlach <CAM.Gerlach@Gerlach.CAM>

Doc/library/threading.rst

@@ -598,14 +598,25 @@ and "recursion level" in addition to the locked/unlocked state used by primitive
 locks.  In the locked state, some thread owns the lock; in the unlocked state,
 no thread owns it.
 
-To lock the lock, a thread calls its :meth:`~RLock.acquire` method; this
-returns once the thread owns the lock.  To unlock the lock, a thread calls
-its :meth:`~Lock.release` method. :meth:`~Lock.acquire`/:meth:`~Lock.release`
-call pairs may be nested; only the final :meth:`~Lock.release` (the
-:meth:`~Lock.release` of the outermost pair) resets the lock to unlocked and
-allows another thread blocked in :meth:`~Lock.acquire` to proceed.
+Threads call a lock's :meth:`~RLock.acquire` method to lock it,
+and its :meth:`~Lock.release` method to unlock it.
 
-Reentrant locks also support the :ref:`context management protocol <with-locks>`.
+.. note::
+
+  Reentrant locks support the :ref:`context management protocol <with-locks>`,
+  so it is recommended to use :keyword:`with` instead of manually calling
+  :meth:`~RLock.acquire` and :meth:`~RLock.release`
+  to handle acquiring and releasing the lock for a block of code.
+
+RLock's :meth:`~RLock.acquire`/:meth:`~RLock.release` call pairs may be nested,
+unlike Lock's :meth:`~Lock.acquire`/:meth:`~Lock.release`. Only the final
+:meth:`~RLock.release` (the :meth:`~Lock.release` of the outermost pair) resets
+the lock to an unlocked state and allows another thread blocked in
+:meth:`~RLock.acquire` to proceed.
+
+:meth:`~RLock.acquire`/:meth:`~RLock.release` must be used in pairs: each acquire
+must have a release in the thread that has acquired the lock. Failing to
+call release as many times the lock has been acquired can lead to deadlock.
 
 
 .. class:: RLock()
@@ -624,25 +635,41 @@ Reentrant locks also support the :ref:`context management protocol <with-locks>`
 
       Acquire a lock, blocking or non-blocking.
 
-      When invoked without arguments: if this thread already owns the lock, increment
-      the recursion level by one, and return immediately.  Otherwise, if another
-      thread owns the lock, block until the lock is unlocked.  Once the lock is
-      unlocked (not owned by any thread), then grab ownership, set the recursion level
-      to one, and return.  If more than one thread is blocked waiting until the lock
-      is unlocked, only one at a time will be able to grab ownership of the lock.
-      There is no return value in this case.
+      .. seealso::
 
-      When invoked with the *blocking* argument set to ``True``, do the same thing as when
-      called without arguments, and return ``True``.
+         :ref:`Using RLock as a context manager <with-locks>`
+            Recommended over manual :meth:`!acquire` and :meth:`release` calls
+            whenever practical.
 
-      When invoked with the *blocking* argument set to ``False``, do not block.  If a call
-      without an argument would block, return ``False`` immediately; otherwise, do the
-      same thing as when called without arguments, and return ``True``.
 
-      When invoked with the floating-point *timeout* argument set to a positive
-      value, block for at most the number of seconds specified by *timeout*
-      and as long as the lock cannot be acquired.  Return ``True`` if the lock has
-      been acquired, ``False`` if the timeout has elapsed.
+      When invoked with the *blocking* argument set to ``True`` (the default):
+
+         * If no thread owns the lock, acquire the lock and return immediately.
+
+         * If another thread owns the lock, block until we are able to acquire
+           lock, or *timeout*, if set to a positive float value.
+
+         * If the same thread owns the lock, acquire the lock again, and
+           return immediately. This is the difference between :class:`Lock` and
+           :class:`!RLock`; :class:`Lock` handles this case the same as the previous,
+           blocking until the lock can be acquired.
+
+      When invoked with the *blocking* argument set to ``False``:
+
+         * If no thread owns the lock, acquire the lock and return immediately.
+
+         * If another thread owns the lock, return immediately.
+
+         * If the same thread owns the lock, acquire the lock again and return
+           immediately.
+
+      In all cases, if the thread was able to acquire the lock, return ``True``.
+      If the thread was unable to acquire the lock (i.e. if not blocking or
+      the timeout was reached) return ``False``.
+
+      If called multiple times, failing to call :meth:`~RLock.release` as many times
+      may lead to deadlock. Consider using :class:`!RLock` as a context manager rather than
+      calling acquire/release directly.
 
       .. versionchanged:: 3.2
          The *timeout* parameter is new.
@@ -658,7 +685,7 @@ Reentrant locks also support the :ref:`context management protocol <with-locks>`
 
       Only call this method when the calling thread owns the lock. A
       :exc:`RuntimeError` is raised if this method is called when the lock is
-      unlocked.
+      not acquired.
 
       There is no return value.