I have compared side by side 10x42mm vs 10x50mm binos and, all other things being equal, there seem to be no significant difference, just very minor differences, in my opinion. But there is quite a bit of difference in weight & size, ie the 42mm bino is slightly lighter and smaller, which is important if you have to carry and use the bino for a few hours of birding or astronomy.would a 50mm show significant difference over a 40mm?
There are more differences when comparing magnification. All other things being equal, higher magnification allows object to be more visible because image scale is bigger and the background sky is darker (higher contrast due to smaller exit pupil). However there is a tradeoff for binos as higher magnification means smaller true field of view (TFOV) and less steady image if handhold (and with a unsteady image, you see less detail). So if you are getting your 1st bino to star-hop and learn the constellation, I believe the 8x40 would be better. If you have steady hands and want a bino to observe binocular deep sky objects, the 10x50 would be more suitable. But you won't go wrong with either choice.
The Nikon Action EX series seems to be good value for its price range, and it has good reviews. I have checked out the Nikon Acton EX 8x40 which is waterproofed, wide field and with good eye-relief for those with glasses. However it is not sharp-to-the edge (which is expected if you are paying just $200) and the lens coatings are not the best (it is also not fully multicoated). But it should be good value for beginners. Do note that the Nikon Action series are cheaper, not waterproofed and not the same quality as the Nikon Action EX (ex for extreme).