I don't often use 打输, I would say it's not actually a compound phrase. I may use it when talking about video games I guess... 这局打输了 I lost this match 打败 is the general term to mean defeat/defeated 被对手打败了/我打败了对手

打輸 is less severe than 打敗. It's like losing a battle (competition etc) vs losing the war

A打輸B → B win A打敗B → A win A被打敗 → A lose A被B打敗 → B win A被打輸 → X (wrong usage)

打输 means lose the game 打败 means win the game

打输应该是两个词，输修饰打，是结果，打败是一个词