Appendix:Proto-Indo-European/ātr-

We use lower-case h when we know which laryngeal it is, and upper-case H when we don't (h₁, h₂ and h₃, vs. H). It's inconsistent to mix an upper-case form like Hʷ with lower-case forms like hₓ: the paper you linked to seems to consistently use upper case for everything, so case doesn't mean anything in that context, and doesn't have to be followed. As for the *Hʷ notation itself, the problem is that it makes it look like it could be any laryngeal, but with rounding added, rather than a specific unknown sound that has the property of sometimes causing adjacent vowels to become rounded. I would convert your to notation to ours like this:

• hₓ = H
• Hʷ = h₃

Thus, *Hʷet- would be our h₃et-

Thank you for this introduction. So which reconstruction would you most prefer then? *h₂eh₁ter-, *h₂eh₂ter-, *h₂eHtr-, *HeHtr-, *HeHt(e)r-, *Hḕt-, *h₃et˖r- or *h₃et-?

I'm more familiar with the conventions used by our PIE entries than I am with the intricacies of laryngeal theory itself. I'll let CodeCat answer that