Is the word zakai (exempt) (as opposed to hiav, guilty) different from the word mutar (permitted) or the word patur (actually a phrase, patur aval assur, without penalty but prohibited)? If so, how are these terms used differently, and when is the word zakai used rather than these other terms. (I noticed the word zakai on page 4A of TB brachot).
Zakai refers to a person who wins a court case; mutar refers to something (or someone) being permitted; patur means exempt from an obligation.
Zakai means innocent or worthy. It refers to a person who is either innocent or guilty; good or bad.
Mutar means permissible. It refers to an act, which can be permitted or forbidden.
Patur means exempt or discharged. Like zakai, it refers to the person, but it does not speak to the character of the person as zakai does; it simply discharges the person from liability or obligation.
The Jastrow Dictionary is a useful source.