[JimD]

You posted several pictures but it's still hard to see what is going on. Are you sure there is no oil on the back of the engine? I'm not sure it's coming from the transmission but perhaps that's because I can see what you see. In any event, if it's full, then you haven't lost much. If it isn't full, then you need to fill it. I used, and like, Red Line MTL but many people use something thinner - automatic transmission fluid. It doesn't take much to change and fill it.

If the oil starts on the engine I would not suspect the manual transmission. I would especially not suspect the manual if I could find fluid in the fill hole (i.e. it is basically full). Most cars with a few miles on them show signs of oil on the back of the engine and the transmission. It isn't necessarily leaks. It can be spills from an oil change.

The key parameter is whether your transmission is full of fluid. If it isn't you need to get it full and monitor it. If it is full, I would clean off the transmission and check it again in a month or so.

Jim