水平 VS 程度 in the context of language proficiency
I think there are differences regarding the terminology usage from different regions.

To me, 我的中文水平不夠 sounds more natural. I guess 我中文程度不够 could be more natural to folks from Taiwan. If you add a modifier for 程度: 我中文理解程度不够, then it would be more natural to me。

水平 means 'level' and it's typically measured by 高（high） and 低（low）. For example, you could say 我中文水平不高, 我中文水平比较低.

程度 means 'extent', it's typically measured by 深（deep） and 浅(shallow) . For example, 我对中文理解程度不深, 我对中文理解的很浅.

There are also some other alternatives to express 我中文水准不够: 我中文水平不行， 我中文不行，我中文水准不行， 我中文水准不够, 我中文不够水准.