Thanks to everyone who posted links. From them I busted a couple misconceptions I had about projections and gathered this quick summary.
It should be mentioned that many projections don’t truly preserve any attribute. Their intent is typically to minimize all types of distortion thereby not eliminating it in any specific property. Jack of all trades, master of none sort of idea. Also I want to mention most GIS environments group CRS by properties in one way or another. This can give a pretty good clue as to what you’re dealing with and what other systems it may be similar to. So it’s probably a good idea to start there.
Now on to answer the question, ‘Is there a place in the CRS name that indicates this?’ Yes, and no. Sometimes. If it is in the name it’s probably one (or more) of these fancy cartographer words.
An Album of Map Projections, John P. Snyder & Phillip M. Voxland
If it’s not in the name, it’s probably just the guy’s name who came up with it. Projections like Robinson, or Eckert, or even Mercator come to mind. In these cases a quick internet search will likely get you what you need. And probably before too long you’ll be familiar with at least a few of them to have a general idea at first glance.
There’s also a bunch of other fancy cartographer names for different things (like aspect), which give you other information about the projection but are not as explicit about distortion as the ones listed above.
As for the example of Alberta 10-TM (Forest). In this case distortion isn’t explicitly mentioned and this is where prior knowledge or the aforementioned internet search will come in handy. Like others pointed out, UTM projections are good at angles and are pretty useful when the area of interest has a primarily north-south extent – much like Alberta.