If you have only one wedge, make some wooden wedges on site. Use an axe to cut a few 8" sticks of wood and sharpen them to a pointed wedge. If you use only one wedge, it can and will become stuck at some point, with no way to retrieve it...
Looks like a nice pile of staves! Perhaps it would have been better to ask questions first, and then start splitting later, but this amount of staves should be fine for now. The only thing that has not yet been addressed, is "so we began the split in a place that would give us 3 equally sized staves". I think this is where you went wrong. That will mean the smaller piece will be more flexible and will be 'peeled off' the bigger piece. The small piece will have grain tear out on the far end and might become too narrow.
Instead, try this:
- Split the half log into two pieces, one measuring 2/5 and the other 3/5. The difference between these two pieces is so small, that will split without grain tear out. Then split the 3/5 piece in two equal parts, so you end up with one 2/5 and two 1½/5 pieces. Splitting in equal parts is always easier, so halves, quarters or eigths, but some logs are not big enough to split them into eigths.
- For three equal parts, you can also split the half log in the middle of the length. Don't start at the end, but drive a sharp axe into the middle, from the bark side towards the heart of the tree. Gradually progress the splits towards both ends. Don't drive one axe all the way down, but rather more carefully. Keep the split with a small opening and go towards the ends as quickly as possible. I hope that makes sense