You're correct, they basically mean the same thing. As you mentioned, the only difference is that outside *my* window is more specific: outside the window = no information regarding possession. outside my window = outside the window + the window is my window. He likes to look outside the window, but outside my window, he can't see anything. When my cat is indoors, he likes to watch the birds from outside the window. (There are one or more windows in the house, and the cat watches the birds from one or more of these windows). When my cat is indoors, he likes to watch the birds from my windows. (There may be one or more windows in the house. One of those windows is MY window, and it's that specific window that my cat watches birds through).

Ai, remember that Rule number 1 is that there are no stupid questions! If you are talking about watching from inside your own house then you would say 'outside my window', but that does suggest that the dogs are quite close to the window and they are walking right outside of it. In this case 'outside my window' is the location where the dogs are walking. If you just wanted to say that when you look out the window you can see dogs walking - perhaps in a park across the road, you would probably say 'I can see dogs walking from my window'. In this case, the window is the means through which you are able to see the dogs, not the location of their walking. If you were in a place other than your own home or somewhere you wee staying (such as a hotel), you would say 'the window' rather than 'my window'.

Also, say we are talking at my home. All the windows are my property. However I would just say, "the cat watches the dogs outside the window". There is no need to specify that all the windows are mine, unless I'm bragging or unless I'm differentiating between one window that I exclusively use all the time. An example of exclusive use would be my bedroom window. A further illustration of the latter point is that there may be one particular window near the cat's bed. Even though it's my property, the cat is the primary user and we might refer to it as his. "He's watching the birds outside his window." "Outside the window" would be perfectly acceptable here as well.