Corner sliding window lock

I'm afraid I don't know what you're asking for, exactly. Did you want a new style mortise lock? If so, I don't think that will be possible. Your door is designed to use a single hook mortise; yours specifically looks like our 82-218. Cutting the hole to accept something larger would be a huge project and probably not worth the effort.

The reason I say that is because double hook mortise locks aren't really more secure. They work just as well as any other, honestly. If there is any scenario where a single hook mortise would simply fall down, then it would be due to a faulty spring inside the mechanism itself; that same fault can happen to a double hook mortise just as easily. 

If I'm understanding your post correctly, you're basically looking to cut open a precisely sized hole in your door to use a lock that would offer only minimal additional benefit. I just don't think it's worth it. 

If I were you, I would stick with the mortise you have now. These are very, very common and work perfectly well in securing patio glass doors. If you have any doubts, you could look into an exterior security feature like the 80-006.

If you want your mortise to activate by pushing down on the lever, instead of up, then you could probably just flip the mortise and keeper upside down.

The mortise on the door you're showing here resembles our 82-265. 

Yes, the keeper is listed on the 82-265 store page under "Related Item(s)" as the 82-249. It is sold separately.

The handle you have on your door now should work fine with it. Any handle that works with the 82-218 can work with the 82-265. They work pretty much the same way.