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I recently read about hydrogen bonding. One thing about which I am really confused is how to distinguish between hydrogen bond acceptors and donors.

I looked it up on SE and found this post, which said that water can form four hydrogen bonds as it has 2 hydrogen bond acceptors (the H atoms) and two H bond donors (the two lone pair of electrons on O atom).

When I looked it up on Wikipedia, it said:

In a hydrogen bond, the electronegative atom not covalently attached to the hydrogen is named proton acceptor, whereas the one covalently bound to the hydrogen is named the proton donor.

And both of them seem to be contradictory. Which is correct?

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Dissenter's answer which you linked appears to be wrong (as of now). I have left a comment.

The conventional way in which we talk about hydrogen bond acceptors/donors is: in a hydrogen bond $\ce{X-H...Y}$, the hydrogen bond donor is $\ce{X-H}$ and the hydrogen bond acceptor is $\ce{Y}$. (See for example Chem LibreTexts, or this other question on hydrogen bonds.)

Of course, water is capable of acting as both an acceptor and as a donor:

Hydrogen bonding in water

I wouldn't call it a "proton donor" like Wikipedia does, though, as that suggests some kind of acid-base reaction occurring, like

$$\ce{\color{red}{H2O} + \color{blue}{H2O} <=> \color{red}{OH-} + \color{blue}{H3O+}}$$

In this reaction the red water molecule is acting as a proton donor (Brønsted acid) and the blue water molecule is acting as a proton acceptor (Brønsted base). But these terms should only be used in the context of a proton transfer, and not hydrogen bonding.

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I would say it is a bit of a misnomer, saying that something is accepted or donated. A hydrogen bond is an attraction between a very positively polarised proton and a well defined lone pair, from N, O or F. When this attraction is broken the molecules separate, no protons are donated or received (overall, of course proton exchange can and does occur)

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