Only apply select_on_focus in Input if the focus event wasn't produced from an app focus event.

src/textual/app.py

@@ -4052,7 +4052,9 @@ class App(Generic[ReturnType], DOMNode):
                     # ...settle focus back on that widget.
                     # Don't scroll the newly focused widget, as this can be quite jarring
                     self.screen.set_focus(
-                        self._last_focused_on_app_blur, scroll_visible=False
+                        self._last_focused_on_app_blur,
+                        scroll_visible=False,
+                        from_app_focus=True,
                     )
             except NoScreen:
                 pass

src/textual/events.py

@@ -736,6 +736,10 @@ class Focus(Event, bubble=False):
         self.from_app_focus = from_app_focus
         super().__init__()
 
+    def __rich_repr__(self) -> rich.repr.Result:
+        yield from super().__rich_repr__()
+        yield "from_app_focus", self.from_app_focus
+
 
 class Blur(Event, bubble=False):
     """Sent when a widget is blurred (un-focussed).

src/textual/widgets/_input.py

@@ -641,7 +641,7 @@ class Input(ScrollView):
 
     def _on_focus(self, event: Focus) -> None:
         self._restart_blink()
-        if self.select_on_focus:
+        if self.select_on_focus and not event.from_app_focus:
             self.selection = Selection(0, len(self.value))
         self.app.cursor_position = self.cursor_screen_offset
         self._suggestion = ""