Here's a patent illustration for the Estes model rocket motor.
There has been an argument regarding whether it should be called a engine or a motor. On this patent it is called a "Toy Rocket Motor".
This patent was filed July 24, 1959 and approved March 24, 1964.
I don't quite understand the Figure 1 drawing. It looks more like a bottle rocket. Maybe it was to show a comparison between a fireworks bottle rocket and a model rocket.All drawn engines look to have a fuse in their nozzles.
Take a look at this early Carlisle patent.
In Figure 5: While it looks like a model rocket motor, there is a fuse outside the motor casing acting as a delay. It runs from the top of the propellant to the ejection charge.
From what I've read, some of the RDC (Irv Wait) engines had a fuse outside of the casing. The length of the fuse determined how long the delay was before ejection.