How many types of MagSafe are there?
According to Apple, there are three types of MagSafe adapters:


"T" Style 
"L" Style
MagSafe 2


The "T" and "L" style are the original MagSafe (or MagSafe 1, if you will) and are compatible with each other.



As for identifying the adapter to your specific model, Apple also provides an excellent support document:  Find the right power adapter and cord for your Mac notebook

As for the MagSafe being dead, according to articles written just today about a patent filed by Apple, they may be bringing it back as an adapter (MagSafe to USB-C)