The Undeath word of power is somewhat similar to Animate Dead, but the latter mentions creating variants (e.g. fast zombies and bloody skeletons) and the former doesn't.
Rules as written, is there a way to create variants as a word caster?
RAW, no. The Undeath word just says "raise as skeletons or zombies." It does not say "as animate dead" or have any other verbiage allowing a RAW variant. Variants say they may be created by animate dead but not other things kinda like it.
Of course it's completely reasonable to do so, but if you're strict about RAW then you're on the wrong side of it.
The spell word text is not clear about mentioning that variants are not permitted. The Animate Dead spell also says nothing about variants (except in notes of the editor, regarding rules about interaction between variants and the spell found in the variant monster rules, not the spell). Also, in the same page, you can see there is a lesser version of Animate Dead that specifically forbid from summoning variants. As fast zombies are still zombies and bloody skeletons are still skeletons, and the spell do not forbid summoning their variants (when there is a similar spell that have that especific limitation) I do not see why,by the rules, you would not be able to summon them using the Undeath word as long you respect the HD limits imposed by the spell.
That said, a DM allowing this may still see reasonable adding house rules imposing restrictions for summoning variants, as they are usually more powerful than the normal ones. Those you mentioned has already their own when summoned with Animated Dead:
You could try to port these restrictions to the words of power system as requiring word combinations for specific variants (not that easy, because as mentioned in the comments, often it is hard or plain impossible to find exact equivalences for certain spells in this system). You can also go the easy route and simply applying a HD modifier for the summoning, maybe based on the extra CR these variants usually have.