Yes, the rulings that come up during game play become rules for that house. The difference between house rule and home brew as I use it and have seen many other use it, is that house rules deal with mechanics. Home brew does not. For example. Saying elves in my game are burly and all get +2 to strength is a house rule. Saying that orcs are the noble protectors of humankind and that only orcs have the necessary purity to be Paladins is home brew.
I agree that the game can't be played without house rules. It can be played without home brew, though.
I don't see it that way (though I often use them interchangeably). I see it almost the opposite. A house rule is a particular rule interpretation of change to an existing rule. Homebrew could include house rules, but also includes new rules, different settings, fundamental changes to the structure of the game, etc. House rule to me is more specific and homebrew is broader and sometimes deeper in scope.
I play a homebrew version of 5e using 3 defenses; AC as DR; HP & BHP, etc. These are mechanical changes, but go way beyond house rules in my opinion - so I would call it homebrew.
I don't see where that only deals with trained people, though. Proficiency isn't added to the character's ability score. It's added to the roll. That rule would result in success or failure for trained and untrained equally, assuming identical ability scores.
Okay. That's all dealing with the rule that says that you only roll when the outcome is in doubt, though. What we have been discussing is whether setting two different DCs for the same task, or narrating two different results for an identical end roll depending on trained or untrained, are house rules. Those optional rules just let the DM decide when success is automatic and you don't need to roll.
The question I was responding to was with regarding whether or not a DM could, by RAW, adjudicate that two different characters who have the same roll (in total) get a different result for the check. With the rules I quoted you can do that as follows:
Option #1: DC 16
Character A has ability score 22, rolls a 15, but auto succeeds because the DC is more than 5 below his/her ability score
Character B has an ability score of 18, rolls a 15, but fails the check
Option #2: DC 15
Character C, rolls a 14, fails the check
Character D, rolls a 14, passes the check because they have proficiency and are 11th level
These both give examples of how two characters can have the same roll, but get different results. They are not as flexible as what you want, but as guidelines they point in a direction that allows you to adjudicate similar cases more along the lines you are looking for.
Not that you need RAW support to do what you want - so I'm not sure what your arguing for or about. Are you just trying to prove that you have a better understanding of the rules?