Comma after introductory phrase followed by a verb

The website you consulted apparently wanted to craft a rule in as few words as possible. It didn’t occur to whoever wrote it that in such mundane sentences such as

With a hand mixer, beat on low speed for 1 minute. — Recipe, Conzatti’s Italian Market.

a verb immediately follows on the heels of a prepositional phrase.

Actually, all would be well were “except in imperatives” added to the rule, but to show you how it all works, we have to go the long way around.

Not all prepositional phrases in initial sentence position are alike. Those crucial to making a sentence grammatical will trigger a subject-verb inversion, a remnant of a Germanic past where, similar to German and Dutch, the verb must come in second position in independent clauses and the subject follows.

Your example sentence follows a construction known as a directive inversion: where a verb of directed motion takes a prepositional phrase as a complement, which has been fronted to initial sentence position:

Into the raging river plummeted the raft with its frightened occupants.

This is the inverted form of:

The raft plummeted into the raging river with its frightened occupants.

Now the raging river has no occupants, but a reader might begin parsing the sentence as if the with-phrase modified river instead of raft. The inversion is not only more dramatic; it’s clearer.

While the prepositional phrase with its frightened occupants is certainly of importance to those in the boat, it is not to the sentence itself:

The raft plummeted into the raging river.

Leave the river out, and you get:

The raft plummeted with its frightened occupants.

And your first question is where? The sentence makes no grammatical sense without the phrase.

In imperatives, however, even prepositional complements in initial sentence position take a comma before the verb:

Into a large mixing bowl, put the carrot, cabbage, beetroot, onion, chilli and half of the chopped coriander. Recipe, mofoodblog.com

Perhaps it’s the understood subject you that triggers the comma.

There are two other flavors of inversion beyond the directive relevant to your question: a locative inversion, where a location complements the verb, and one with with or by, which requires the adverb only or solely. One could, I suppose, term this construction an instrumental inversion since it describes the means by which some action or state occurs, but I don’t think anyone has bothered giving it a name.

The main thing, of course, is that you recognize the inversion and when you construct sentences yourself following this pattern, you don’t throw in a comma where it doesn’t belong.

Here’s another directive inversion like your example:

Down the slope from the wind-swept summit into the valley rode the posse of Jake Bush. — Arthur Murray Chisholm, The Land of Strong Men, 1919, 409.

A locative inversion:

Atop an incline at the end of the street stood the U.S. Capitol, the broad stairs sweeping upward to the colonnade and capped by the elaborate three-tiered dome. — Greil Marcus, The Shape of Things to Come: Prophecy and the American Voice, 2007.

And one each with only with and only by:

Only with eyes closed could he see the iridescent scowl, white screen of brow that opened like the Hudson at twilight, where he stood breathing in the open air and shedding each day's labors. — Jacqueline A. Kolosov, Jacqueline McLean, Danish Ocean, 2003.

From all angles of government, we get the same message: “It's not our fault, and only by raising taxes can we solve the problem.” — San Bernardino County Sun (CA), 13 Feb.1991,9.

Sentences like these are what the rule has in mind, not imperatives. Perhaps you should email the website and suggest adding “except imperatives” to save others the confusion you faced trying to apply the rule.