The answer by @EdM is very good. Here are some more general remarks. There is no "objective" notion of what constitutes a "true outlier". As argued by Davies and Gather (1993), it makes sense to define "true outliers" relative to a reference distribution for the non-outlying observations. There then needs to be a decision what the reference distribution should be. The normal distribution is extremely popular for this, due to the fact that (a) it will not regularly generate observations that intuitively seem "outlying", (b) it comes up in statistics all the time, often connected to the Central Limit Theorem (but also because the ubiquitously used mean is Maximum Likelihood for the normal distribution), and this makes many people believe that all kinds of data that occur in reality are at least approximately normally distributed. This notion is somewhat problematic as no statistical theory dictates that any real data should indeed be (approximately) normally distributed, or that the normal distribution shape in any way indicates that observations are "good" and trustworthy.

In a real situation it would be best if the user could specify a reference class of distributions for modelling the non-outliers such as normal or Poisson distribution, but unfortunately in many applications, also apparently here, there is no strong background information on which such a choice could be based.

Even If you had such a reference class of distributions for the non-outlying observations, you'd still need to specify a threshold above which observations are classified "outliers", and there are no general principles that would determine such a threshold (I read somewhere that the famous factor $h=1.5$ in the boxplot stems from John Tukey deciding, based on some experience but without any specific reproducible rationale, that "2 is too big and 1 is not big enough" - the nice comment by @NickCox provides more information on this).

Davies and Gather proposed to choose the threshold dependent on the dataset size $n$ so that the probability to identify at least one outlier in a dataset of size $n$ drawn i.i.d. from the reference distribution is smaller or equal than some specified small probability $\beta$ (e.g., 0.05). Applying this to the boxplot and assuming the normal distribution as reference one obtains, as long as $n$ is not very small, a factor quite a bit larger than 1.5, and consequently fewer observations will be identified as outliers than the standard boxplot identifies.

In case the data distribution is systematically skew even for non-outliers, theory assuming the normal as reference distribution will no longer be appropriate. The adjusted boxplot as cited in the question is a nice tool supposedly not assuming a specific reference shape, but in order to give a statistical basis for choosing a threshold, once more one would need to specify a reference distribution. Note in particular that (as for the standard boxplot) the threshold of 1.5 may well be too low (and may identify too many outliers) in situations in which (a) the distribution of non-outliers has heavy tails (which means that there can be non-outliers in quite some distance from the center, relative to non-extreme quantiles of the distribution) and (b) outliers of relevance (indicating "fraud", erroneous observations and the like) are known to be quite extreme and rare.

Transforming skew data to look more symmetric and then applying techniques based on normality is not a particularly bad way of dealing with skewness when it comes to outlier identification. But it doesn't solve the problem that there is no guarantee that after such a transformation normality is the "correct" reference for the non-outliers. Furthermore, outlier identification methods based on p-values (you haven't explained what exactly you have in mind) do rely on the normality assumption in a stronger way than methods such as the boxplot or the identifiers proposed in Davies and Gather (1993; the most prominent one being based on median and MAD). So even after transformation, reference distribution and threshold choice are somewhat subjective and would need to rely on user decisions.

Note that generally statistical theory has no authority over how ("good" or not so good) real data should look like, and this means that there is no unique statistically "correct" or "valid" way how to tell apart outliers from non-outliers based on the data alone without involving background knowledge about what constitutes an outlier. There needs to be user input regarding the characteristics of outlying and non-outlying data, and the aim of outlier identification (regarding more or less generous thresholds in particular).

Practically both the boxplot adjusted for skewness and transforming to more symmetry, then using normality-based outlier identification are not wrong, but unfortunately statistical theory (or experience) will not conclusively tell you which approach is best for your situation. If possible you can manually check observations from more to less extreme to see whether they qualify as fraud or suspicion of fraud. If you need something automatic, one approach is to look at a number of datasets manually, think about the situation, and based on this choose a suitable h-threshold larger than 1.5 with the adjusted boxplot. Without more precise background knowledge I'd expect this to be about as good as it gets.

PS: One thing that you may be able to decide from background knowledge, and which is important with skew distributions in particular, is whether you want to be able to identify outliers based on being too large and being too small, or whether only too large (or only too small) observations are relevant as potential fraud. Once more, no statistical theory and no data on their own can tell you this, but note that standard normality-based outlier identification (be it boxplot, median/MAD or outlier tests) will identify outliers on both sides, which may not be appropriate (changing this is of course easy in principle, but it has impact on the issue of threshold choice).

Davies, L., & Gather, U. (1993). The Identification of Multiple Outliers. Journal of the American Statistical Association, 88(423), 782–792. https://doi.org/10.1080/01621459.1993.10476339