Any ideas why running a point buffer (50 and 150 meters) would result in this shape?
ArcGIS 10.2 - All projections match
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So the reference work for understanding what has happened here is Understanding Coordinate Management in the Geodatabase.
Essentially, the fact that "projections match" is of no consequence, because a coordinate reference in ArcGIS encompasses more than just the coordinate system (of which, a projected coordinate system (PCS) is one option -- the other is a geographic coordinate system (GCS)).
In Esri parlance, a coordinate reference (CR) includes:
The origins are used to establish a lower-left corner (or lower bound) for integerization, and the scale factor is used to determine the partitioning of the discrete number coordinate values per map unit (To make things even more complicated, Desktop presents the scale factor as its inverse (resolution) in some contexts, but not all). The cluster tolerance is used by ArcGIS for snapping (it defaults to twice the scale). Note that these rules are applied in the X/Y, Z, and M dimensions.
Your problem is in the realm of the XY scale factor being too small. Looking at the image with grid lines enabled, it's easier to see how the coordinates are being snapped to a coarse grid (~10m interval).
The default resolution in a PCS is usually 1mm, so you've used a value 1000 times (or at least 100 times) too coarse for the use case. Unfortunately, CRs are a permanent data characteristic of an enterprise or file geodatabase feature classes, so the only way to "alter" one is to drop the table and start over (from the original data source).