You are right. If you are talking about something attached in some way to a flat surface, whether horizontal or vertical, the usual preposition is 'on'. For example, 'There's a fly on the window' or 'There's a dirty mark on the window". 'In the window' is a little different. If we talk about something being 'in' a window, we are thinking of a window as being a display area. For example, goods are put on display in a 'shop window', which is essentially a glass cabinet. If you are using a window for display purposes, you can use 'in', even if what you are displaying - in this case a sign - is fixed to the surface of the glass.

Thank you!

If it is BEHIND the glass it is "In the window" because you look through the glass to see it If it is IN FRONT OF the glass and ATTACHED TO THE GLASS it is "On the window" because it has to be attached to the window. So both can be used but it depends where it physically is in relation to the glass!