This catch is known as the Lamb's Tail Tube catch. Without the strip of metal and the ball of solder it is just a tube catch.
This is known as a T-pin and you can see why. It resembles a capital T. Widths vary but 1/2 inch wide is common.
This is also a T-pin however it is much more narrow. Commonly called the N-type or Narrow T-pin.
Spring Pin with cover plate. Badge makers often stamped their hallmark into the cover plate.
A common pin catch that resembles the letter C is aptly called the C catch.
