The relevant portion of the linked article is sloppily written.
It is correct to say that "whom" is ungrammatical in these sentences:
He is demanding £5,000 from the elderly woman who has ruined his life.
Mr Reynolds is highly critical of journalists, who just use labels to describe him.
It is incorrect to describe "elderly woman" and "journalists" as "grammatical subjects".
The noun phrase "the elderly woman" is the object of the preposition "from", and the antecedent of the relative pronoun "who", which is what actually acts as the subject of the relative clause "who has ruined his life".
The noun phrase "journalists" is the object of the preposition "of", and the antecedent of the relative pronoun "who", which is what actually acts as the subject of the relative clause "who just use labels to describe him".
The grammatical role of the antecedent is not the same as the grammatical role of the relative pronoun, and so each is assigned a different case.
In sentences like this, you shouldn't match the case of a relative pronoun to its antecedent; consider that we wouldn't match the case of a non-relative pronoun like they/them to its antecedent. We would say "Mr Reynolds is highly critical of journalists. They just use labels to describe him," not *"Them just use labels to describe him."
"...from the elderly woman whom he claims has..."
According to many respected grammarians, the article is incorrect (although repeating a common prescriptivist peeve) when it describes the use of whom in the following sentences as "wrong" or "a mistake":
He is demanding £5,000 from the elderly woman whom he claims has ruined his life.
Mr Reynolds is highly critical of journalists, whom he says just use labels to describe him.
Here, whom stands for the subject of a clause embedded within the relative clause. In the first sentence, the relative clause (with "claims") has the subject "he", and the further embedded clause (with "has ruined..") has the subject "who(m)". Because the relative pronoun is the subject of the most embedded clause, "traditional grammar" calls for the use of "who", but the fact that it is not the subject of the relative clause seems to make "whom" possible here for at least some English speakers.
From what I remember, Huddleston and Pullum's CaGEL describes the use of "whom" in such contexts as so established in the usage of educated adults that it should be considered grammatical (for speakers who use it). Here is a Language Log post by Arnold Zwicky that makes this point: "ISOC, ESOC". Admittedly, this is a rather technical point; if you are uninterested in the study of descriptive grammar and just want to learn the pattern of usage that is most likely to be considered "correct" by others, it's safer to go with who in sentences like these.
This construction is the topic of an older question on this site, The use of nominative "whom" (as in “persons whom it is foreseeable are likely to...”) See user F.E.'s comments below the question post for more references to descriptive treatments of the use of whom in this context.