As you already know convergent evolution is when two separate lineages evolve the same (or similar) trait, and this is generally assumed to be a result of similar selection. For example, if flying helps both birds and bats catch insects to eat, then convergent evolution of flight is somewhat unsurprising. Other examples include those of mimicry where the convergence on a common colour patterning reduces predation rates.
The wikipedia page on convergent evolution does a pretty good job of explaining the subtle difference between parallel and convergent with its illustration. In convergent evolution, two lineages have dissimilar starting points and arrive independently at similar end points. In parallel evolution, two lineages have similar start points and arrive independently at similar end points.
"...a convergent change at an amino acid site refers to changes from different ancestral amino acids to the same descendant amino acid along independent evolutionary lineages ... It is distinguished from a parallel change, in which amino acid changes along independent lineages have occurred from the same ancestral amino acid (see fig. 1A for examples). Both convergent and parallel evolution, if verified, suggest adaptive evolution."
This, and other sources (page 113), suggests that convergent and parallel evolution/change/mutation are relevant terms. In your context you are looking for a term to differentiate two cases.
Case 1: convergent evolution of a trait, by different mutations; either different nucleotides at the same site, or mutations at different sites. This is seen in your lactase example.
Case 2: convergent evolution of a trait, by the same mutations; the same nucleotide changes at the same site(s). This is seen in your white fur example.
A key point is that your are looking at convergence at two different levels, phenotypic and molecular. In both cases you have convergent evolution at the phenotypic trait level. The cases differ at the molecular level where you have convergent and non-convergent mutation respectively. I have found one term being used in this paper.
"In some cases it has been shown that different loci are involved in phenotypic convergence (e.g. Refs [8,25,26]), demonstrating that similar phenotypes can be reached through alterations of distinct enzymes. However, other studies have traced phenotypic convergence to modifications of homologous genes (e.g. Refs [3,5,6,26,27]); in this paper such phenomena will be further referred to as convergent recruitment (Glossary)."
I think the term you are looking you are looking for would be convergent recruitment to describe case 2, and some form of antonym* for case 1, perhaps you could say non-convergent recruitment. However, non-convergent could make it sound like no convergence is occurring so I'm not a fan and you should be careful in any papers to clarify. I'd say something like:
"Phenotypic convergence by the recruitment of different genes and/or mutations is herein referred to as non-convergent recruitment, in this case convergent evolution is occurring but through non-homologous changes at the molecular level."
You would then have to be consistent in your writing, talking about the phenotypic level as (non-)convergent evolution and molecular level as (non-)convergent recruitment. E.g.
"The convergent evolution of flight occurred as a result of non-convergent recruitment."
* I couldn't find an established antonym term, and as I suggest in the post, I don't think non-convergent recruitment is a perfect antonym, we could discuss it in chat or the comments to try to come up with some new terminology. Perhaps "non-homologous convergent evolution" or "non-genetic convergence" would work? In this paper they simply refer to it as "convergence through different mutations".