I always hear good things about GPower, but have never used it.

Every program should provide its formula and they should all be comparable - so I don't think there is a best. You probably run into issues with the defaults settings and many options available, plus I am unsure if sample size calculations will be different between trying to power a likelihood ratio or Pearson chi-square. I just run those both and found a difference of 61 versus 62 subjects per group. Also you need to state a reference group many times.

I ran your numbers in SAS and got the following:

The POWER Procedure

Pearson Chi-square Test for Two Proportions

Fixed Scenario Elements Distribution Asymptotic normal Method Normal approximation Group 1 Proportion 0.7

Group 2 Proportion 0.9

Nominal Power 0.8

Number of Sides 2

Null Proportion Difference 0

Alpha 0.05

Group Actual Power 0.803

N per Group 62

I believe simulation could give you more variability in the number generates if the seed is not set - typically using the system clock. To be cautious, don't forget to add on an additional 10% or so if you have any doubts about the sample you may get in comparison to the numbers used to generate the calculations.