Thank you for the ref to the AIC Myths and Understandings! Regarding the point on normal on $log(x_i)$ vs lognormal on $x_i$, I am fitting using the lnorm functions provided in poweRlaw which build on base R's dlnorm and plnorm functions, so this is correctly fitting the lognormal I guess! Both models are being compared over the same support $x_i \in [x_{min}, Inf)$. So then we are in good shape to directly compare the AIC?

I also saw your answer here stats.stackexchange.com/questions/111693/… but do not fully understand part 3. What do you mean by treat the estimated parameters as the population value and then compute the KS statistic as it simplifies the computation? Can we not simply compute the ks statistic between the simulated data and estimated parameters of the simulated data?

Good point. In alternative 2, for each iteration we estimate the parameters of the target distribution in from the simulated data. This is in line with Greg Snow's answer here: stats.stackexchange.com/questions/45033/… and what is outlined in Clauset et al. 2009 (save for their x_min section). As I understand it, the idea is because we don't know the parameters, so we should replicate the estimation procedure to somehow incorporate the parameter estimation error into the KS statistic distribution.

So, if I knew somehow that all lakes above a certain size (xmin) were produced by process A, and that this process resulted in a lognormal distribution, I would test my hypothesis by fitting a lognormal to those values. Here, instead, we truncate the left side and assume that our process operated on the entire range of values, but the sample contains no data from any values below xmin. What's the reasoning behind this? I may be misinterpreting the meaning or significance of xmin.

I modified the question to clarify what I was actually asking. Is fitting a truncated log-normal, rather than a log-normal to the entire set of values above xmin, the correct way to go?