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Example:

A tall(,) green pole.

This site says that cumulative adjectives don't need comma.

In my example, "tall" describes shape, and "green" describes color. So they are cumulative adjectives.

But this site says that "tall" and "green" are equivalent and coordinate adjectives ...

I'm confused. Should there be a comma or not?

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    Yes, cumulative adjectives don't take commas. And in your first example, "tall" and "green" are not cumulative, but coordinate adjectives. They have equal weight. The pole is tall, and it is green. You can easily reverse their position and it would still make sense. – AIQ Jun 28 at 5:31
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    @AIQ No, that's not right. They easily can be cumulative. If you have a series of green poles, and half are tall while the other half are short, then you have tall [green poles] and short [green polls]. You can't tell if they are coordinate or cumulative in this case simply by looking at the adjectives. The only way to know is from context and intended meaning. – Jason Bassford Jun 28 at 5:53
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    @JasonBassford But OP said "A tall ..." and then I said "in your first example ..." So it seems there is just one pole here, which is tall. – AIQ Jun 28 at 5:56
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    @AIQ There could be 100 poles. 99 of them are short. You point at the one that isn't, and say, "Look! A tall green pole." No comma. And even if there is only a single pole, it depends on how you are interpreting it. – Jason Bassford Jun 28 at 5:58
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    @JasonBassford Ahh ... I see, thanks for clarifying. – AIQ Jun 28 at 6:03
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I was asked to turn my comments under the question into an answer.

In this case, what's being described can either be a tall green pole or it can be a tall, green pole.

It depends on how the pole is being interpreted and described.


Cumulative adjectives

With this kind of adjectival use, one modifies another that follows it. They are cumulative because each adjective modifies the adjectival phrase that follows.

Say you are standing in a field of poles, and all of them are green. Also, every pole you've seen so far has been short.

Suddenly, you spot a green pole that happens to be tall:

"Look! It's a tall green pole.

In this case, the items being described are considered to be "green poles", and tall serves to modify that thing. In other words, you have this:

tall green pole

tall [green pole]
→ "That green pole is tall."


Coordinate adjectives

With this type of adjectival use, each adjective modifies the noun independently of anything else.

At a different time, you could be walking down the street and see a pole. You observe that it's tall. You also observe that it's green.

As far as you're concerned, the fact that it's tall has equal important to the fact that it's green—they are two separate items of interest.

You might say this:

"Look! It's a tall pole. Oh, and it's a green pole too."

→ "Look! It's a tall pole, and it's a green pole."
→ "Look! It's a tall and green pole."

→ "Look! It's a tall, green pole."


This method of replacing the conjunction (and) with a comma is the same method that's normally used in other expressions:

"I ate apples and I ate oranges and I ate pears."
→ "I ate apples, oranges, and pears."

The comma stands in for and, as well as serving to help break up the list items visually. (In speech, it might also indicate a slight pause between the items.)

The one thing that's different about normal expressions is that we use an and before the final list item. That is not the case with coordinate adjectives.


Getting back to the pole, in this case, there is a "pole" that is both tall and green:

tall, green pole
→ tall, green [pole]
→ "That pole is both tall and green."

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[1] A tall, green pole.

[2] A tall green pole.

Both are possible, though with a slight difference in meaning. Punctuation distinguishes coordination from the stacking of modifiers.

In [1] "pole" is modified by a coordination of adjectives, giving the meaning "pole that is both green and tall".

In [2], by contrast, there are two layers of modification: "pole" is modified by "green" to form the nominal "green pole", and this in turn is modified by "tall", allowing a somewhat different interpretation, i.e. "green pole that is tall".

In most situations, the difference is likely to be of little consequence, but if you were writing about an event where a number of green poles of a certain height were being erected, the difference may be important. In speech, there may be emphasis on "tall" in [2].

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Might any green, tall pole suffice?

It shouldn't.  To the average native speaker of my acquaintance, it wouldn't.  A green, tall pole is generally objectionable.  The occurrence of that word order is comparatively rare, and might exist primarily due to irrelevant happenstance.  For example, we don't want to count those times when "a green tall building" represents an environmentally-friendly skyscraper.

Your second citation provides a hint of sound logic with a faulty premise:

For example, in “a tall, green pole,” a comma separates the two adjectives because they are coordinate, or equivalent. To test this fact, either replace the comma with and (“a tall and green pole”) or reverse the order of the adjectives (“a green, tall pole”). If these changes make sense (even though the original syntax is better), the adjectives are coordinate, meaning that they both refer to the pole, and the comma is required. In “a weathered green pole,” by contrast, the adjectives are noncoordinate: Weathered and green do not each modify pole; weathered modifies “green pole,” so no comma is necessary.

Replacing the comma with a conjunction is too large a change.  Doing so makes the adjectives explicitly coordinate, regardless of whether they would represent a coordination inherently.  The coordination there is governed by something external to the adjectives themselves.  That isn't where the hint of sound logic lies.

In contrast, reversing the order is a sensible test . If we wonder whether a dense, warm fog is as suitable as a warm, dense fog, we can find evidence to support the notion.  The frequencies of those orderings occur within an order of magnitude.  That small a difference can be explained as a mere matter of style and preference.  Dense and warm seem inherently coordinate.

The faulty premise is that "a green, tall pole" makes sense in the absence of some exceptional context.  We can engineer such a context, surely, but we don't assume such a context out of the blue.

 
The royal order of adjectives is a lie.  The underlying truth is closer to a simple progression from general to specific applicability.  We can find circumstances under which size should be a more tightly bound attribute than color, or when size and color have equal applicability.  It is, however, a useful, productive and instructive lie.  From it, we can expect color to be more tightly bound.

In the general case, we can assume that "tall" ought to modify "green pole".  In the general case, a tall green pole makes more sense without the comma.  In regard to your second citation, the "green, tall pole" lacks a foundation and doesn't stand up to scrutiny.  The pole falls down.  It doesn't provide enough support for abandoning the royal order.  Size remains something outside of color.

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