Does this rule exist, based on usage?
That's an awfully loaded question, so for a loaded answer we can ask Google for an NGram:
As you can see, "reoccurs" is very uncommon compared with "recurs", when the rule would suggest we should see it more of the time. We get much the same picture graphing "reoccur" against "recur". This tends to suggest that we would be on very dodgy ground basing this rule on usage.
Does it mean the rule is wrong? Well, no. Consider:
Also, according to this distinction, is the first recurrence of something strictly a reoccurence and not a recurrence? That seems a little incredible.
What you are missing (and what doesn't show up in the NGrams) is that the first reoccurence is also a recurrence. Two data points do make a (trivial) repeated pattern, in exactly the same way that you can draw a straight line through any two points on a graph.
Notice also that something that recurs only perhaps happens at regular intervals. As you said, this is a matter of nuanced meaning, not hard and fast rule.
In summary, in theory we could use recurs and reoccurs to draw the distinction between regular and irregular repetition, but in practice we don't use reoccurs often enough for the distinction to be obvious to readers.