If a published paper had an author ordering [A,B] where A and B are primary authors and the ordering is alphabetical is it acceptable for B to list on his website the ordering [B,A] so his first-authorship would look more obvious to people skimming his website?
In Can co-first authors’ positions be swapped in a CV?, the answer with the most upvotes says
No. The reference is the reference, changing it to move yourself higher could be misinterpreted as willful deceit.
I believe the same logic applies to a website.
However, in a website, you do have more freedom in style and formatting than you would in a CV (although you still don't have the freedom to change author order...) For example, if you want to call attention to first authorship, you could list first authors and co-first authors in a slightly different weight or different color than the other authors.