What's proper to count in home's sq ft?

There is no hard fast cut and dry rule for all scenarios;
there are different rules for different reasons... for instance the MLS does not allow an agent to list below grade sq ft of finished space in the MLS "sqft" field.. If it was not originally finished when the home was built and built to the same standards as original home construction .. then it is a finished basement and not included in sq ft counts on the MLS.  Then some counties have other rules that say that if the finish was done to code and with permits and has windows and fire exits and is livable with hvac and such.. then it can be included in sqft totals... but it is called livable or usable space..  and appraisers adjust for this space when looking at comps. 
then some homes that are split levels always include the lower level finish in total sq ftg.. .so it would depend on style of home too..  
the way to tell if the sq ft includes the lower level is usually by looking at the assessors records and see what the main floor and upper floor dwelling space is listed to be.. then you should know..  or ask the agent or seller.. or a final method would be to engage a space planner or appraiser to measure the home.  once you visit a lot of homes you can get a good feeling for sqft miscalculations... usually.
hope that helps .

Usually, it's the living space that is not only finished, but is heated/cooled and inhabited.

As such, garages aren't counted unless they've been converted into living space (and hopefully, done with permits).

Basements won't count, unless they're finished, heated, cooled, vented, etc. and conforms to building codes and defined as habitable.

Check with your local building codes for descriptions of habitable space in a dwelling.

Or...you can also get your own appraiser who will give you the rationale about which areas are counted in the square footage.

Good luck!

Dear Skia.   Recently I attended a seminar where a certified appraiser told us that banks want to see finished living area that has HVAC that is above grade.   Does a finished, "daylight" basement count?   Depends on where you are and who is doing the appraisal.   I would contact an agent in your area for customary practice.
Good luck.