Performance Name, Slot Name, Rig Name, Amp Name, Cabinet Name, other tags... are all different animals. People can use these for different purposes.
Performance Name could reflect the name of a song.
Slot Name could reflect a musical Scene e.g. "Solo".
Rig Name could reflect a combination of information like "Gilmourish SM57".
Amp Name could be "Hiwatt 100".
These are just examples.
Rig Name is normally a combination of pieces of information about provider of the Rig, amplifier, amp settings, microphone, instrument, purpose of the Rig.... It can be edited when you store a RIG (Store/Rename) or in the Rig Settings under Rig Tags (also Rig Tags in Rig Manager).
As long as Slot Name equals Rig Name, both change, if another Rig gets loaded into a Slot. Slot Name follows Rig Name. As soon as Slot Name and Rig Name deviate, the Slot Name sticks and doesn't follow, if you load another Rig. Slot Name can be changed in Edit/Slot Settings or Rig Manager.
Amplifier Name is one of a whole bunch of amplifier tags, which can be edited in the amplifier module under Amp Tags. The same applies to Cabinet Tags.