Having spent a bit of time on this, I'm going to post it anyway, although KitFox just beat me to it!
No; the "new attribution rules" apply specifically to quoted material. A simple link to further reading does not require attribution, but where there is material reproduced from elsewhere it is not sufficient merely to provide a link.
A comment has reminded me that "quoted material" need not always be textual. The rules apply to text and images.
Even if quoted material is licensed in such a way that attribution is not required, it's still better to say where it came from, as those who check these things won't necessarily know the licence terms and SE rules require attribution regardless.
The test for acceptability
Only direct quotations need a citation. A poster's own words do not.
If just the visible content of the post can be reproduced without losing the source of a direct quotation, then the citation is acceptable.
If losing any links means that all reference to the source is also lost, then an explicit reference needs to be added.
Worked examples
Acceptable:
The Columbia Guide to Standard American English (1993) confirms that surveil is a late addition to English:
Surveil is a fairly recent back-formation, a verb describing what a surveillant ("watcher," an even more recent coinage) does. All these terms are Standard, although some might consider surveil and surveillant primarily police and spy argot.
This is acceptable because the source of the quote is cited in plain text. SE rules say that it should also be a link, and indeed it is in Steven Yarg's answer where I grabbed this from. [Did you see what I did there?] Some material is not available online and therefore cannot be linked to.
Also acceptable:
The Columbia Guide to Standard American English (1993) confirms that surveil is a late addition to English.
This is acceptable because there is no quotation. The sentence is entirely the poster's original words. In this particular example, some corroboration such as a quote or a link would be very useful, but there is no quotation here to cite.
Also acceptable:
This is what the ESL people call the second conditional. The first conditional is used for something which we expect will happen, though it might not.
Again, this is acceptable because there is no direct quotation. The link is simply a link to further reading on the matter. The two sentences I've quoted here come from the original answer which OP Colin Fine is asking about.
Not acceptable:
reference : The action of mentioning or alluding to something.
This is not acceptable because the source of the quote is not cited in plain text. It doesn't matter if the headword here is a link; the source — which is ODO in this case — must appear in plain text.
Also not acceptable:
reference : The action of mentioning or alluding to something.
Source
Again the source of the quote is not cited in plain text. If the link is removed, all that is left is the word "Source". In this case, the text [Source]
in the link could usefully be replaced with [ODO]
to identify the source and provide a tidy link.
Meta: if it's obvious where material has been quoted from, perhaps because there is a link to it, then by all means edit the post to include the source in plain text. But there's no need to do that if there is no actual quotation; and if it's not obvious then all it is possible to do is comment. When editing a post or including a citation, try to make it look pretty: a tidied title which forms the text for a link is far better than a bald URL.