There is not much difference between the methods other than that the number of trials in method 1 is more predictable than in method 2. Consider the following example:
Say we tried 5 simulations using method 2 and the results are:
2, 2, 1, 4, 1
So our estimate of p = 1/mean = .5
This also means that we have run 10 trials:
0, 1
0, 1
1
0, 0, 0, 1
1
We can see this as a method 1 simulation and see 5 successes out of 10 trials, which corresponds to a estimate of p of .5
This is no coincidense. For method 2 our estimate of $p$ is 1/mean, which is $\frac{N} {\sum_i k_i}$ (whereby $k_i$ are the number of trials till success). We also know that this represents a method 1 simulation with $\sum_i k_i$ trials and $N$ successes, so our estimate will again be $\frac{N} {\sum_i k_i}$. So there is no real difference between method 1 and method 2 other than that the cost of method 1 is more predictable than the cost of method 2. As long as you compare like with like, that is, compare a method 2 simulation with a method 1 simulation with $\sum_i k_i$ trials, you will find no difference in the results with respect to the point estimates. As you use different variance functions with gamma and logit families you might get slight differences in standard errors. In Stata I would inspect this like so:
clear all
set obs 100
gen t = _n <= 50
gen p = .1 + .3*t
gen u1 = runiform() < p
local i = 1
sum u1, meanonly
local stop = r(mean) == 1
while `stop' == 0 {
local j = `i'
local i = `i' + 1
gen u`i' = runiform() < p if u`j' == 0
sum u`i', meanonly
local stop = r(mean) == 1
}
egen k = rownonmiss(u*)
glm k t, family(gamma) link(power -1)
margins, over(t) predict(xb)
gen id = _n
reshape long u, i(id) j(iter)
glm u t, family(binomial) link(logit)
margins, over(t)
The relevant outputs are:
for the gamma model:
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
t |
0 | .0988142 .0112884 8.75 0.000 .0766895 .120939
1 | .462963 .052888 8.75 0.000 .3593043 .5666216
------------------------------------------------------------------------------
for the logit model:
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
t |
0 | .0988142 .0132661 7.45 0.000 .0728132 .1248152
1 | .462963 .0479803 9.65 0.000 .3689232 .5570027
------------------------------------------------------------------------------
So the predicted $p$ are exactly the same and there are slight differences in the standard error.