When attempting to translate, "He remained silent, as if in thought," I said
Er blieb still, als wenn er im Denken war.
and I was corrected with
Er blieb still, als wenn er am Denken war.
I find both "im Denken" and "am Denken" in DWDS. What is the difference and why the correction?
Denk mir Kaffee und Schokoladendonuts herbei, wenn du schon am Denken bist.
Frage: Die Menschen suchen nach Kontinuität, wollen aber frei im Denken sein.
Both your sentences are missing a verb ;)
In general, constructions like "am (Verb) sein" are a progressive form mostly used in colloquial speech. It was known as the "rheinische Verlaufform" (roughly "Rhinelandian progressive form"), but recently the less localized term am-Progressiv is used more often.
The idea of the construction is roughly similar to something like "to be currently at (verb)-ing":
Sie ist am Lesen.
She's at reading.
or, in more natural English
She's reading at the moment.
She's busy reading.
Other examples would be
Was möchtest Du essen? - Ich bin noch am Überlegen.
What would you like to eat? - I'm still thinking.
Die Stimmung war am Kochen.
The mood was boiling / at the boiling point.
So, one of your sentences in question needs a form of "sein":
Er ist am Denken.
And in Konjunktiv it would be
Er blieb still, als wenn er am Denken wäre.
Er blieb still, als wenn er am Denken sei.
The difference between "wäre" and "sei" in this case would probably warrant a question of its own.
On the other hand, "in etwas sein" just means "to be in something". So, "Er ist im Denken" would be something like "he's in thinking", which could be understood as similar to "he's in thought". But it wouldn't be phrased like that in German. The example you found at DWDS uses a construction similar to "free in thinking" or "free at thought".
So, of the sentences you suggested, only
Er blieb still, als wenn er am Denken wäre.
would be correct.
But as mentioned before, the am-Progressiv is a quite colloquial construction. So there's a bit of a styles clash between the two parts of your sentence, as they are from different registers. A better way of phrasing, as Sixpence already suggested, could be
Er blieb still, als wenn er in Gedanken wäre.
Er blieb still, als wenn er in Gedanken versunken wäre.
"In Gedanken versunken" ("to be sunk into thought") is a fixed phrase that means something like "to be absorbed in thought".
"Im Denken" is just wrong in this case.
"Am Denken" is possible, but rather colloquial. It refers, however, to the act of thinking.
"He remained silent, as if in thought": Er blieb still, als sei er in Gedanken (versunken).
The important difference betwee the correct example "Die Menschen ... wollen ... frei im Denken sein" and your wrong example "... als wenn er im Denken [wäre]" is the word frei. With the preposition "in" you specify in which regard you are free.
Other examples:
In these examples the word "in" does not refer to the moment you are actually doing something.
Regarding the progressive form:
As a native speaker born in Berlin, I would use "beim Denken" instead of "am Denken", but I would use this form only in slightly different context. In this specific case I agree with Henning Kockerbeck's answer that "in Gedanken versunken" would fit much better.
Er blieb still, als wenn er am Denken war.
This is a very colloquial way to put it, but it would be understood as "He remained silent, as if he was thinking". It technically is wrong and it sounds uneducated (apologies if this is a regional dialect somewhere).
Er blieb still, als wenn er im Denken war.
This is just wrong. "Im Denken" is neither a correct nor a common way to express a progressive action. The example you give
wollen aber frei im Denken sein.
does not use "im Denken" as a progressive. In this sentence "das Denken" means the discipline of thinking in general. Similar you could say "Das Denken fällt mir schwer" ("thinking is difficult to me").
In both cases is "Denken" is closer to the English "thinking" and not a good translation for "thought", which is better translated as "Gedanken".
Er blieb still, als sei er in Gedanken.
You are falling for a common error in translation, and that is translating word by word, verbatim.
Conventionally, in written Standard High German, the idea behind your English example sentence would be expressed using a different construction. The one that comes to my mind immediately is:
Er schwieg, als würde er nachdenken.
That is, both er blieb still as well as als wäre er am Denken are uncommon in written Standard High Germand and stilistically awkward.
Im Denken seems unlogical, as a person (as a physical being) commonly cannot be inside thought (compare als wäre er im Haus with als wäre er im Denken). In thought is a figure of speech in English which does not exist in German.
In spoken vernacular, constructions with am [Tun] are common, though: "Als ich gekommen bin, war er grad am Arbeiten."
