Recur vs. Reoccur

Is there any difference between the verbs reoccur and recur?

Several sources suggest that they are synonymous, but some fine-tuners suggest that there is a nuanced difference, such as Grammarist, which explains

Something that recurs happens repeatedly, perhaps at regular intervals. Something that reoccurs happens again, but not necessarily repeatedly or at regular intervals.

Does this rule exist, based on usage?

Also, according to this distinction, is the first recurrence of something strictly a reoccurence and not a recurrence? That seems a little incredible.

Your quote from Grammarist matches what I would have answered. Expanding on it a little, I would use recur for an event that is expected to happen repeatedly or regularly and reoccur for an event that might happen again, but is not expected to.

The U.S. presidential elections recur every four years, but no one expects voting issues like the Bush vs. Gore election to reoccur anytime soon.

To get the feel of it, try going into the negative:

-- "We don't want this to reoccur!" means that we don't want it to ever happen again.

-- "We don't want this to recur!" means that we don't want it to happen repeatedly at regular intervals, but randomly and sporadically is OK.