Are both expressions okay?

"she has claimed the stake on a name "

I’ve learned a phrase 'stake a claim ’ , and never heard of ‘claim the stake’ . However, if I look at each word separately, ‘claim the stake’ also seems to make sense. But I’m not sure if it is generally acceptable usage. Please give me some explanation.

If you stake a claim, or claim the stake, it means you claim something for your own. In this case, the woman has decided on a baby name and will not allow anybody else in the family to use it.

I have to say that I haven’t really heard “claimed the stake” used very often. I think “to stake a claim” is much more common.