I think you misunderstand what they're doing with the formula. Delta is the size of the difference you want to detect. In this case, they're saying they want to detect a 5% (relative) change in a 5% (absolute) conversion rate. So σ^2 = 0.05 . (1-0.05), but Δ is 5% of 5% or 0.0025 and the denominator needs to be 0.0025^2.
If you go to your reference [2] and enter the numbers 0.05, 80, 0.05, 0.0525, 1.0, you'll see that they come up with a sample size of about 122k in each group (so 244k in both together).
(The figure of 304 or 440 is what you would get if you wanted to detect an absolute change of 5% in the conversion rate: going from 5% to 0% or to 10%.)
If you go to your reference [2] and enter the numbers 0.05, 80, 0.05, 0.0525, 1.0, you'll see that they come up with a sample size of about 122k in each group (so 244k in both together).
(The figure of 304 or 440 is what you would get if you wanted to detect an absolute change of 5% in the conversion rate: going from 5% to 0% or to 10%.)