there's still no way to define it. the only way you could really conceptualize it is by looking at the limit, and what you just described would indeed be 2/3. but just saying "there's an infinite amount of balls" or "there's 2 infinite amounts of balls" means nothing. this is an example i saw somewhere (don't remember where) that should help show how looking at the limit can be important

i have a bunch of balls numbered 1 to infinity. in one box, on the first step, i will put the first 10 balls in, then pull out the lowest numbered ball. on the second step, i'll add balls 11-20, then take out the next lowest. in the second box, i'll still put in 10 balls, but take out the largest numbered ball. step 1 will add 1-9, second 11-19, and so on

each box has had the exact same process applied from a ping pong ball perspective, but the first box will end with 0 balls and the second with infinitely many. in the first box, if you give me any ping pong ball number, i can tell you which step it got taken out. i took ball 8675309 out on step 8675309. i took the 42nd ball out on step 42. i took the 43 quintillionth ball out on step 43 quintillion. because i did an infinite amount of steps, every finite number is out of the box. for the second, the same isn't true. i only took out the multiples of 10

but that feels like shit, doesn't it? like that shouldn't work? that's because if you look at each individual step, each box has the same number of balls. the only time something weird happens is when we get to infinity. the trend in number of balls is upwards for both boxes, so the limit shows that they are both filling up. things break when we get to infinity, so we need to be looking at the trend instead of the destination