Is it possible to get in a variable the GTM tag name? I would like to see in the console the tag name of the tag triggered:
GTM Tag Name inside cHTML tag screenshot

I have found the variable exists on the google_tag_manager object called instance name:

There's the GTM Monitoring API in public beta. You could use it to get fired tag names. I doubt the object you referenced has the actual log of fired tags. More like the list of all tags in the container. Monitoring API is probably the closest you can get to the fired tag names easily.
This article from Simo could be handy too. He's using Big Query as a destination in his example, but it's easy to modify to whatever.