Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:<p>

**Theorem 4.32.** Equivalent are:<br>
<ol>
<li>Every lattice has a maximal filter.
</li>
<li>Every complete lattice has a maximal filter.
</li>
<li>Every distributive lattice has a maximal filter.
</li>
<li>Every closed lattice has a maximal filter.
</li>
<li>AC.
</li>
</ol>
<p>

In this theorem a <i>closed lattice</i> is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).
<p>

Herrlich, Horst<br>
[Axiom of choice][1] <br>
Lecture Notes in Mathematics, 1876.<br> 
Springer-Verlag, Berlin, 2006.<p>

Herrlich credits the equivalence of Item 3 and Item 5 to<p>

G. Klimowsky. <br>
[El Theorema de Zorn y la existencia de filtros e ideales
maximales en los reticulados distributivos.][2]
<br> 
Rev. Union Mat. Argentina, 18:160-164, 1958.<p>

Where Herrlich writes in terms of AC and in terms of the existence of a maximal filter, Klimowsky writes that Zorn's Lemma is equivalent to <i>En todo reticulado distributivo con primer elemento, todo filtro está contenido en un ultrafiltro.</i>


  [1]: https://link.springer.com/book/10.1007/11601562
  [2]: https://inmabb.criba.edu.ar/revuma/pdf/v18n4/p160-164.pdf