diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 5259bbc38c2f22..ecdef214bbdd1a 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -8,9 +8,9 @@
     Callable,
     FrozenSet,
     Hashable,
+    List,
     Optional,
     Union,
-    List,
 )
 import warnings
 
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index f25426e4f8786c..52b4d3251d517c 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -339,7 +339,7 @@ def argmax(self, axis=None, skipna=True, *args, **kwargs):
     # Rendering Methods
 
     def format(
-        self, name: bool = False, formatter=None, date_format=None, na_rep="NaT"
+        self, name: bool = False, formatter=None, na_rep="NaT", date_format=None
     ) -> List[str]:
         """
         Render a string representation of the Index.
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 018b6806c5603d..0878b9344a2f1c 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -1231,13 +1231,17 @@ def _format_native_types(self, na_rep="nan", **kwargs):
 
     def format(
         self,
+        name=None,
+        formatter=None,
+        na_rep=None,
+        names=False,
         space=2,
         sparsify=None,
         adjoin=True,
-        names=False,
-        na_rep=None,
-        formatter=None,
     ) -> list:
+        if name is not None:
+            names = name
+
         if len(self) == 0:
             return []
 
@@ -1265,13 +1269,13 @@ def format(
             stringified_levels.append(formatted)
 
         result_levels = []
-        for lev, name in zip(stringified_levels, self.names):
+        for lev, lev_name in zip(stringified_levels, self.names):
             level = []
 
             if names:
                 level.append(
-                    pprint_thing(name, escape_chars=("\t", "\r", "\n"))
-                    if name is not None
+                    pprint_thing(lev_name, escape_chars=("\t", "\r", "\n"))
+                    if lev_name is not None
                     else ""
                 )
 
diff --git a/pandas/io/formats/format.py b/pandas/io/formats/format.py
index 22cdd8e235e0b0..b5d5f930f881d1 100644
--- a/pandas/io/formats/format.py
+++ b/pandas/io/formats/format.py
@@ -330,9 +330,8 @@ def _get_footer(self) -> str:
 
     def _get_formatted_index(self) -> Tuple[List[str], bool]:
         index = self.tr_series.index
-        is_multi = isinstance(index, MultiIndex)
 
-        if is_multi:
+        if isinstance(index, MultiIndex):
             have_header = any(name for name in index.names)
             fmt_index = index.format(names=True)
         else:
diff --git a/pandas/io/formats/html.py b/pandas/io/formats/html.py
index 7ea2417ceb24b4..13f0ab1e8a52ce 100644
--- a/pandas/io/formats/html.py
+++ b/pandas/io/formats/html.py
@@ -442,6 +442,7 @@ def _write_hierarchical_rows(
         frame = self.fmt.tr_frame
         nrows = len(frame)
 
+        assert isinstance(frame.index, MultiIndex)
         idx_values = frame.index.format(sparsify=False, adjoin=False, names=False)
         idx_values = list(zip(*idx_values))