I wholeheartedly agree with Kawshik .
You are better off , using the smaller ( so to speak ) bore which you are comfortable with , than the larger bore which you might struggle to shoot accurately . Use only good quality soft point bullets of the heaviest weight possible , in your 9.3 × 62 Mauser ( which , in our time in India , we used to call the .366 Mauser ) . Avoid solid metal covered bullets at all costs for buffalo . From personal experience , l can attest that a .366 Mauser solid metal covered 286 grain bullet makes a very small hole in a water buffalo or gaur's heart and it could sometimes go up to 2 kilometers without showing any signs of slowing down , which was very challenging for us professional shikarees .
A good double lung shot , with a soft point bullet ( 286 grains is about right ) fired from a broad side position , can make short work of a gaur or Indian water buffalo .
Were l personally given a choice of rifle to use , to go after buffaloes , my personal choice would be a .450 / 400 Nitro Express using 400 grain soft point bullets .
Even though l have no experience with the .404 Jeffery calibre , an internet search indicates that it is ballistically identical to the .450 / 400 Nitro Express .
Therefore , either of these 2 would be my personal choice .
In your .416 Remington magnum calibre rifle , l do not recommend using 300 grain bullets for buffalo , because the natural weight of .416 Remington magnum calibre bullets is 400 grains . A lighter bullet in a larger bore , reduces sectional density and therefore penetration
( Though to what degree in practicality , l cannot say ).
That said , if you could handle the 400 grain bullets in your .416 Remington magnum , then l would recommend it straight away. With the .366 Mauser , shot placement is a little more critical .
All the best.