Misfire is NO ignition of the powder.
Hangfire is DELAYED ignition.
Squib is a light or underpowered ignition.
A misfire CAN be a hangfire, and that's where the wait comes in. When I teach, and on my range, the shooter is required to wait, holding the rifle securely downrange, for a MINIMUM of 60 seconds after a misfire to see if there is a hangfire situation.
I recall one instance 2 summers ago of a hangfire that MIGHT have been a disaster- semi-experienced shooter (modern stuff only- first time ML) has a misfire, waits for the required minute, places the rifle in the rack to be checked by instructor (me), takes off eye + ear protection, reaches for the field rod to put into the bbl to signify a loaded rifle, and it goes off. Maybe a total of 2 minutes from trigger pull to discharge. Don't know why it took so long to fire, same load we had been using all day- 60gr pyrodex, prb, #11 remington cap.