That's like crediting a football team's chapionship win 100% to the players and 0% to the coach. It's true in the most literal sense, yet widely missing the mark.

Congress does all legislation. Saying the President "just" signs legislation handily relieves the President of responsibility for any laws they sign during their tenure.

But Presidents typically take credit for their legislative wins, so we also give them credit for budgets they sign.

(It is also the case that one measure of the efficacy of an administration is the quality of the budgets they are able to coax out of Congress.)

Eh, the majority party in controls the budget negotiations, and the president is either the leader of that party or of the minority party. Either way there is considerably more involvement than just signing the result of other people’s work.

And contextually, the change in the deficit still aligns with the Party of the President (and party of the senate majority)

Reagan had to make a lot of concessions to the Democrat majority in Congress to get the budget passed. It wasn't much of the budget he wanted.