he was confident that they could discover how to cheat death together. It's not an assertion that Anakin could gain the power with Palpatine's help.
Yes, it is. Together means with Palpatine's help. Those mean the same thing.
the first is a claim that he already knows enough to use it on his own.
No, he did not claim to be able to use it on his own.
Palpatine starts off by saying "let me train you in the dark side and YOU will be able to save your wife." He doesn't say "I can do it all by myself, so I will do this thing FOR you." Why not?
Instead, he says "THROUGH me YOU will achieve a power." That refers to them working together.
Later he says "if we work together we can discover the secret."
These positions are entirely consistent. It is a consistent narrative.