diff --git a/pytensor/compile/function/types.py b/pytensor/compile/function/types.py
index 9cc85f3d24..14ffa223c8 100644
--- a/pytensor/compile/function/types.py
+++ b/pytensor/compile/function/types.py
@@ -949,32 +949,27 @@ def __call__(self, *args, output_subset=None, **kwargs):
 
                     except Exception as e:
                         i = input_storage.index(arg_container)
-                        function_name = "pytensor function"
-                        argument_name = "argument"
-                        if self.name:
-                            function_name += ' with name "' + self.name + '"'
-                        if hasattr(arg, "name") and arg.name:
-                            argument_name += ' with name "' + arg.name + '"'
-                        where = get_variable_trace_string(self.maker.inputs[i].variable)
-                        if len(e.args) == 1:
-                            e.args = (
-                                "Bad input "
-                                + argument_name
-                                + " to "
-                                + function_name
-                                + f" at index {int(i)} (0-based). {where}"
-                                + e.args[0],
+                        function_name = (
+                            "pytensor function"
+                            if self.name
+                            else f"pytensor function '{self.name}'"
+                        )
+                        argument_name = (
+                            f"argument {arg.name}"
+                            if hasattr(arg, "name")
+                            else "argument"
+                        )
+                        where = (
+                            ""
+                            if config.exception_verbosity == "low"
+                            else get_variable_trace_string(
+                                self.maker.inputs[i].variable
                             )
-                        else:
-                            e.args = (
-                                "Bad input "
-                                + argument_name
-                                + " to "
-                                + function_name
-                                + f" at index {int(i)} (0-based). {where}"
-                            ) + e.args
-                        self._restore_defaults()
-                        raise
+                        )
+                        e.add_note(
+                            f"\nInvalid {argument_name} to {function_name} at index {i}.{where}"
+                        )
+                        raise e
                 arg_container.provided += 1
 
         # Set keyword arguments
diff --git a/pytensor/configdefaults.py b/pytensor/configdefaults.py
index ca3c44bf6d..8b4c6c233e 100644
--- a/pytensor/configdefaults.py
+++ b/pytensor/configdefaults.py
@@ -638,15 +638,8 @@ def add_error_and_warning_configvars():
     # on all important apply nodes.
     config.add(
         "exception_verbosity",
-        "If 'low', the text of exceptions will generally refer "
-        "to apply nodes with short names such as "
-        "Elemwise{add_no_inplace}. If 'high', some exceptions "
-        "will also refer to apply nodes with long descriptions "
-        """ like:
-        A. Elemwise{add_no_inplace}
-                B. log_likelihood_v_given_h
-                C. log_likelihood_h""",
-        EnumStr("low", ["high"]),
+        "Verbosity of exceptions generated by PyTensor functions.",
+        EnumStr("low", ["medium", "high"]),
         in_c_key=False,
     )
 
diff --git a/pytensor/link/utils.py b/pytensor/link/utils.py
index 9cbc3838dd..780e4c8cce 100644
--- a/pytensor/link/utils.py
+++ b/pytensor/link/utils.py
@@ -313,6 +313,12 @@ def raise_with_op(
         # print a simple traceback from KeyboardInterrupt
         raise exc_value.with_traceback(exc_trace)
 
+    if verbosity == "low":
+        exc_value.add_note(
+            "\nHINT: Set PyTensor `config.exception_verbosity` to `medium` or `high` for more information about the source of the error."
+        )
+        raise exc_value.with_traceback(exc_trace)
+
     trace = getattr(node.outputs[0].tag, "trace", ())
 
     exc_value.__thunk_trace__ = trace