Yes, it's installed correctly in the photo.
I've never been able to remove, or install the oiler as you describe Andy. I've never had a type I stock, and maybe it'll work that way with one of them.
I've tried it that way when installing just to see if it would work by making a loop on the 'oiler-side' and then inserting the oiler into the loop. Even with original thin early type 'A' ('C-tip') slings, the thickness of the sling wouldn't allow me to push the oiler into the slot (with the sling already around it) . I may be misunderstanding what you describe though.
(Or didn't hold my mouth right) ;)
Regardless, I don't have much of a problem installing slings anymore. Usually just run the sling through from the 'sling-side', put the oiler in the stock recess, then hold the stock with the 'oiler-side' down so it 'floats' a bit so it doesn't bind, and then gently thread the sling end back through. I've only had seven carbines and they all were about the same, but some may not have as much tolerance (?)
The postwar type 'C' ('D-tip') are more difficult because of the larger metal end tip and thicker weave, but originals I've tried aren't usually too difficult -
Aftermarket (fake) slings of both types can be a real pain though! Because of the fatter weave and/or tips used on some of them, there isn't enough clearance to push the sling end back through
.
On the worse aftermarket slings I've been able to always push the end back though enough to be able to use needle-nose pliers to grasp the very end of the tip and pull it through.
* I've heard about people filing the opening out to get more clearance, but I've always been able to pull even the fattest aftermarkets through.
I'd rather scratch the tip a little, or even beat the end of the tip flat as I've heard some people have done than file on a stock. :(