Planar means the distance calculation will be done in two dimensions while Euclidean means the distance will be a straight line. The two terms are related but are not synonyms. Euclidean distances can be calculated in two dimensions, three dimensions, or even more dimensions; but the calculation will always be a straight line between points. Planar distances don't have to be Euclidean but are often assumed to be when no qualifier or adjective further describes the distance. For example, Euclidean and Manhattan (or taxi-cab) distances can both be calculated in a plane, which would make them forms of planar distance, but they represent difference distance measures.
ArcGIS Pro handles both vector and raster data, as well as other types, but different fields of study and different industries historically used the two types of data. Just like anything that is studied or used by different groups, it is common for different groups to use different terms for the same or similar objects or ideas because each group interacts with those objects or ideas from different perspectives. For spatial-based analysis, there are lots of non-Euclidean distances that can be measured based on barriers, cost surfaces, networks, etc...., so knowing whether something is straight-line (i.e., Euclidean) is important and saying "planar" would be ambiguous.
Is there perfect consistency and use of "Euclidean" and "planar" across all of Esri's documentation? Probably not, but what is most important is how the term is defined for a given tool relative to the options/parameters that are available. A vast majority of Esri's pages that use "Euclidean" or "planar" also give a brief definition of what that term means for that tool, and that is the definition that matters for that tool.
