The analysis of Reingard in DMNES is correct (and straightforward, there is no reason to doubt it or propose alternate etymologies).
Reinhard is a distinct name with the well-established name element hard, related to modern English "hard". The male name Reinhard was much more frequent over history than the feminine name Reingard that has an archaic sound nowadays.
Although it is said that the themes in dithematic names can combine freely, there are huge differences in the frequency of the combinations, some names are really frequent, others are rare, and lots of possible combinations are hapaxes or completely unattested.