Just because it's a weed doesn't mean it's not a grass. Grasses are a type of plant. Weeds are any plant that's growing where you don't want it. If you don't like the other grass, it's a weed.

I'm not quite sure what species of grass you're dealing with. Zoysia grass doesn't tend to handle wet soils well, getting it to establish and remain healthy may be difficult to impossible. It might be more sensible to just plant a grass that will handle the wet better than Zoysia. If the grass that showed up doesn't look bad, it might be easier to let it be.

As for removing the other grass, there is no right herbicide. To the best of my knowledge there are no herbicides for killing one species of perennial grass without harming another. Best bet would be to work the ground up and try planting again. Again, I would recommend trying a species better adapted to wet soil, because if you replant zoysia you are likely to end up with the same problem.