Need Some Clarification on Signing Off a Letter [closed]

Can you sign off a letter with the following:

Dear Joe,

I appreciated the cake you sent me.

Thank you,

Bob

But doesn't the above sound like you're thanking yourself (Bob) instead of Joe? "Thank you, Bob"

Another one:

Dear Joe,

See you after the new year.

Merry Christmas,

Bob

Again, doesn't the above sound like you're saying merry Christmas to yourself (Bob) instead of Joe? "Merry Christmas, Bob"

No. You are writing

Merry Christmas,

Bob

not

Merry Christmas, Bob.

The first form follows the conventional form of an epistolatory valediction, and would be read in the same way as

Yours faithfully,

Bob

But even if you had written the second form, all on one line, convention takes over.