Okay ... I am a professional troll, hired by Bike Forums to give grief to new members ... but today is my day off ...
You crashed because you overestimated how much pressure you could put on the edge of the tire before the surface shifted. Better tires would just cost more .... it's like being on ice. No shoe is going to make ice not slippery ... unless you have cleats or blades .... but even skaters fall, and people crash on the most aggressive knobbies or even studded snow-tires. there are limits of adhesion and if you pass them you will fall.
That said .... your rim can probably fit a tires from 1.5 to 2.2 inches quite well .... 37 to 55 mm or so. The only thing you have to get right is diameter ... a 700c rim is the same in diameter as a 29" inch rim (neither are 700 mm or 29" in diameter ... that would be too easy .... ) For a front tire I tend to avoid square-edge designs, but in the rear they are okay. If you ride a lot of really soft terrain wider-spaced knobs are okay, but if you ride more hard-packed trails you would be sacrificing overall grip for extreme grip in extreme circumstances.
The issue, particularly off-road, isn't the tire's rubber---a softer tire won't grip better if the road surface is doing the moving. On pavement a sticky, supple tire will help you corner faster (and will wear faster.) off road, the issue is more tire shape and knobs/tread, and tire pressure ... if your tire is hard it tends to dig in deeper, and if you corner hard on soft soil with a hard tire it will tend to scoop out the soil and dump you low-side.
Not saying that you shouldn't buy new tires. .... but there is no magic solution for going into a corner too fast ... well actually, gravity and friction are the solution ... gravity pulls you off the bike and friction eats your skin.
Anyway ... tomorrow I am back on duty and if I see you post, I will make you miserable .... but for today, you got lucky.
(Just kidding about all that .... the mods get mad if I assume people are smart enough not to take me seriously.)
The bike stuff is pretty accurate, the rest is just noise.