Adverbial modifier with the insertion of comma

I was studying about participles and one site a guy asked the following question:

How would the meaning of the following sentences differ from each other?

1. The beach, located on the far side of the river, is known for its beauty.

2. The beach located on the far side of the river is known for its beauty.

The author of the blog post replies back by saying that that in the first sentence "located" is acting as an adverbial modifier while in the second one it is acting as a "noun modifier" and hence the second sentence is correct.

I can't understand how the insertion of a comma makes the word "located" an adverbial modifier. Kindly help me understand it.

Solution 1:

In English, bracketing commas or their lack around participial phrases, appositives, or relative clauses signal whether the element is

• essential (restrictive), i.e. whether the phrase is essential to making a noun/noun phrase specific and defined, thus becoming an integral part of the noun phrase and not marked with commas, or
• nonessential (nonrestrictive) i.e., providing additional information about an already specified noun. Commas are used to signal the distinction, altering the function of the element.

The beach located on the far side of the river is known for its beauty.

In this sentence, located on the far side of the river answers the question “Which beach?” It means that particular beach and none other. Thus the participial phrase is essential and functions as an adjective: the located-on-the-far-side-of-the-river beach as opposed to some other beach.

The beach, located on the far side of the river, is known for its beauty.

This sentence treats the beach as sufficiently defined: you're only talking about one specific beach. Answering the question “Where is this beach?” is not essential to knowing which beach the speaker is talking about.

It functions adverbially if you understand the sentence as

The beach, which is located on the other side of the river, is known for its beauty.

The participial phrase is an adverbial adjunct one could omit and the question of which beach is still answered.