The Ball Oiler that you showed, second, is usually 6 mm OD, and should be available from most of the Model Enginering suppliers, such as Arc EuroTrade, Warco, RDG etc
The only way of extracting the failed one seems to be to either lever it out, if you can, or to force in a self tapper and use that as a point to be gripped in pliers, or wire cutters to prise it out.
Fitting the replacement is merely a matter of gently tapping into place with a piece of wood between nthe new Ball Oiler and the mallet or hammer.
Howard