Yeah, brace yourself for lots of suggestions and opinions, most of which will disagree and contradict.
You said you wanted the bins for birdwatching. But will you be going to watch birds with them and that's the main objective for going out, or do you want them with you to do some speculative bird watching whilst out doing other stuff?
Myself, I've never seen the point of 8x binoculars. That's me though. I wouldn't go with anything less than 10x. Despite what others say, I reckon 10x gives a wide enough view to find things but also gives you enough magnification that you will be able to pick out details and see enough of the object that you are impressed by what you see. Many birders take a 8x but will team it up with a scope - they use the bins to locate and the scope to give a good view. I assume you're not thinking of a combo like that, needing the bins to be a compromise. Personally, for me, 8x doesn't provide that compromise. All they'll do is taunt you - offering suggestion of seeing something impressive but never delivering. Some talk about hand shake at higher magnification. Well, I've never had trouble holding 10x on target. I suppose if you do have a shaky hand then it might be a factor. But I don't have the shakes and I've never had trouble holding a 10x binocular on target.
As for objective lens? I'm going to go against the grain again. Anything over a 25mm lens will not go in a pouch on your hip, not comfortably anyway. You'll have to spend the day with it around your neck. It's not a question of weight, but it is a question of just being a damn nuisance. So if you are willing to have it around your neck, then just go for a 50mm lens which will give you a better image. That's OK if you are going out with the sole objective of birding because you'll be using them all the time. If you want the bins with you in case you see something interesting and suddenly need them, but will mostly not be using them, then you'll want to put them away but have them to hand if needed. In other words, in a pouch on your hip. For that you'll want a 25/26mm objective lens.
So, if what you want the bins for is to view things whilst out and about, not necessarily for dedicated bird watching, then I would suggest a 10x25mm pair.
If you are thinking of dedicated bird watching then what to go for will depend on where you will be birdwatching - always go for a bigger objective lens (50mm) and the magnification will depend on whether you will be looking at near or far objects (experience shows they will mostly be at distance) - so my dedicated birding bins are 12x50. But I can see why you might choose 10x50 (most of my viewing is done over marshes and heaths - so I want something with a bit of range).
A smaller objective lens or magnification might be preferable in dense terrain, or if teamed up with a scope. But I find that in woods you see fewer birds, not least because you have less time to see them (more cover) and because fewer birds live in dense woods than in more open terrain. So, realistically, you'll be using them in more open terrain than close terrain. Choose accordingly.
About this time last year I bought these:
https://www.amazon.co.uk/PRAKTICA-10-Pioneer-Waterproof-Binoculars-x/dp/B01F67LEFQ
Which is pretty much bang on the same price point as the pair you were looking at.
I always have these on my belt when I go out.
I would instantly buy another pair of anything happened to the pair I currently have.
But brace yourself for umpteen dozen other 'swear by' recommendations.
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