I believe this is the same issue as the difference between phylogeny and taxonomy. Phylogeny is based on some distance metric, be it morphological or genetic, and can inform taxonomic rankings, but is often used to differentiate between specific organisms or genomes. For example, if you wanted to know how similar 10 polio virus genomes were, you would find some measure of distance between them and place them on a phylogenetic tree based on those distances. This is an example of phylogeny that is NOT specifically related to taxonomy.
However, taxonomy is a broader term: it is simply a set of rules for defining organisms (or units, etc.) as being similar or different. There's a large (mostly philosophical) discussion around different kinds of taxonomy (including phylogenetic taxonomy), but that's outside the scope of your question. If you're interested in more of this discussion, the field of systematics has plenty of reading material. Most current uses of taxonomy will be in reference to the standard classification scheme, e.g. Kingdom, Phylum, etc.
Specifically for CheckM, what they mean in lineage (synonym for phylogeny as it is used for CheckM) versus taxonomy is the following:
Phylogeny workflow: use a set of markers that discriminate between phylogenetic groups. It's likely that you can define the tree to be used, and then the markers chosen will be the most discriminating between groupings or placement on the phylogenetic tree. Note that for this approach, you can build a phylogenetic tree (for example) for 10 Mycobacterium genomes and use that to generate markers or place your genome bins on the tree. I haven't personally used CheckM for your task, but if you have genomes with phylogenetic distances and you wish to classify them or discriminate between them, this is likely the workflow you want.
Taxonomy workflow: their use of taxonomy is almost certainly the typical Kingdom Phylum... etc. It looks like they let you either generate a set of taxon-discriminant markers or generate a set of markers for a specific taxonomic group here, such as "all Mycobacteria".
Thank you, Steven. You helped a lot :)