You should look at the output. In the toolbox window click on the results tab at the bottom (and if necessary, uncollapse the Average Nearest Neighbor entry).
The NNI ratio, p value, expected and observed are all reported. You need to interpret the actual statistic and not rely in ESRI's GUI interpretation. A random or uniform distribution would be near zero and as the value increases the process becomes more clustered.
Normally, I loath the ESRI spatial statistics tools but, I have the NNI statistic available in one of my R library and ESRI's results are identical. The z value has nothing to do with the clustering, but rather is a measure of significance. To derive the p-value from the z score you can take a two sided approach and use the cumulative density function (eg.,
2 * pnorm(-abs(z)) ). With a z score of 1.96 the p value would be 0.049 which is sitting right at a 0.05 significance level and can be accepted as a significant result.
In the case of the NNI regular, uniform and independent are interchangeable. This statistic does not indicate if there is a dispersal process and you cannot interpret it in the context of a statistic like Moran's-I which is two tailed thus, indication two types of spatial process with zero being random. The NNI indicates randomly dispersed, as tested against a CSR Poisson process, to clustered. The f-hat and g-hat statistics are similar in nature.
Not to be harsh here but, you need to state an actual hypothesis of the spatial process and then select an appropriate statistic and not go on a fishing expedition based on available ESRI tools, which is becoming all to common. An exploratory spatial analysis is also quite warranted as well. For example, as a starting point, have you looked at the open space characteristics of the data. Also, check whether your data is homogeneous or following an intensity process. These are critical questions to address before trying to fit a point process model.