The difference between "all" and "any" in this context is subtle. Formal logic can't distinguish them. The difference is in the ways these words trigger the imagination. I'll try to give you some idea of the primary mental associations these words have for native speakers.
All for one and one for all
Many recipes include a step like this:
Mix all the ingredients in a bowl.
This is the primary sense of all: gathering every item together, or gathering the whole of something, forming a big mass, and regarding it as a single, homogeneous entity—like what you get if you put all the ingredients into a bowl and mix them up.
The phrase "gather all" appears nine times in the King James Bible. Here's a typical example:
For I will gather all nations against Jerusalem to battle; and the city shall be taken, and the houses rifled, and the women ravished… [Zechariah 14:2]
When you have all nations against Jerusalem, that's a pretty one-sided battle. All the nations presumably act as one, in a coordinated alliance. They form one huge army.
Not any, not even a little
The word "any" suggests just one item, or a small amount. Etymologically, it's related to "one" and "an". Psychologically, it's the opposite of "all", even if they (sometimes) literally mean the same thing when negated. Negating a word does not negate the connotations and associations it brings up. Consider "My shirt was not torn to shreds by the tiger" and notice what you imagine.
From the King James Bible:
And the LORD set a mark upon Cain, lest any finding him should kill him. [Genesis 4:15]
This leads you to imagine a single incident: just one person finding Cain and killing him. Rewording it "…so that all who found him should refrain from killing him" would lead a native speaker to think of the totality of the people who would encounter Cain for the rest of his natural life—a large number of people.
I will not take from a thread even to a shoelatchet, … I will not take any thing that is thine… [Genesis 14:23]
Notice that the word "any" is emphasizing smallness, as it usually does when it's negated. I won't even take a bit of thread from a sandal-thong—not any thing of yours. It could be worded with "all": "All thy property shall be untouched and untaken by my hands." This leads a native speaker to dimly imagine a huge pile of property, enclosed in some border, perhaps behind a fence. Abstaining from "all" literally means the same as abstaining from "any", but it wouldn't make sense to illustrate it with a bit of insignificant thread.
The OALD examples
The magazine was all advertisements.
Notice the meaning of homogeneity. "All" leads the reader to consider the magazine as a whole; the assertion is that the magazine is advertisements through-and-through.
He denied all knowledge of the crime.
This leads the listener to imagine knowledge of "all" the facts of the crime, maybe a little like all the ingredients in a recipe. And he is pushing the whole bowl aside.
"He denied any knowledge of the crime" leads the listener to imagine that he doesn't even have knowledge of a small fact about the crime, not even part of a thread.
Does it make any difference in this example? Not much, I think. The only difference I can think of is that "all" has a stronger connotation of completeness (all, you see) or finality than "any". Denying "any" is perhaps a bit more likely to lead someone to challenge it: to probe for a very tiny bit of knowledge. "Not any? Didn't he at least notice that the warehouse door was wide open?" Denying "all" hints that even if he does have some knowledge, he intends to keep his mouth shut. But the difference is very, very slight (not a rule).
Neither version is archaic. Both are commonplace and prosaic in modern English.
By the way, notice that the OALD is technically mistaken in a couple places. To illustrate all's sense of "the whole number of", it gives:
All horses are animals, but not all animals are horses.
Cars were coming from all directions.
What do you think is the "number" of horses and animals being referred to? And how many directions are there? Four? How about in a parking lot?
In sense 2, with uncountable nouns, to illustrate "the whole amount of", it gives:
All wood tends to shrink.
What is the "amount" of wood being referred to? If you said "All the wood in this boat tends to shrink", that would illustrate the whole amount of a substance referred to by a mass noun. In the example, "all wood" refers to wood in the abstract, and normally means all types of wood. Dictionaries are often wrong. Explaining what words mean, especially little, common words like "all", is very tricky. There's probably no way to define these words that captures their meanings fully and precisely.
As usual, negation in English is tricky. Notice that "not all animals" doesn't mean the same as "not any animal".
Why all and any are determiners
Both "all" and "any" indicate how the following noun is intended to refer to the concept it denotes, just as articles do. "All horses" can refers to horses in the abstract, to state an inherent property of horses, or it can refer to all the specific horses in some concrete situation for purposes of stating a rule: "All horses must be fully rested before the race". "All" is well suited for stating rules and generalizations because of its insinuation of homogeneity. When you speak of "all horses", you are implying that all horses are pretty much the same, at least for purposes of the sentence.
"Any horse" is very similar to "a horse". Just like "all horses", it can refer to all horses in the abstract to state a property or it can refer to all the horses in some situation to state a rule, but it invites you to imagine a specific but arbitrarily chosen one: "Pick any horse. It's going to have a mane." "Any" can also express rules, because of its suggestion of arbitrary choice of just one instance.
However, "all horses like to run" suggests a rigid, exceptionless rule, while "any horse likes to run" suggests that you're describing a typical horse and not insisting that there couldn't be exceptions. Since "any" leads you to imagine just one horse, it suggests that you're choosing one at random, which probably will be typical; there could be weird, unlikely horses that you're not imagining. "All" leads you to mentally gather all horses, past, present, and future, actual and potential, into a homogeneous totality, favoring a more rigid interpretation. There's no difference between the literal, dictionary meanings, but by now you know that the dictionary definitions are only part of the story.
With count nouns, "all" goes with the plural and "any" with the singular. As you can imagine, the grammatical details get complex and messy if you try to spell them all out explicitly.