So
@Slats432 and I go back and forth because he's a "club" hockey coach and I'm a "club" hockey parent, both in Edmonton (I hope our banter is all friendly!).
So this terminology might only be to Edmonton/Alberta. "Club" hockey is the elite AA/AAA hockey, where you have to try out and you might not make it. It' still a non-profit club that has existed for decades. "Federation" hockey is the hockey where everyone plays, it's just a matter of what tier you're assigned to. It's also non-profit.
"Academy" hockey is where you go to a school which is also your hockey team (see CSSHL). It's outside of Hockey Canada's control, but they do place a lot of kids in the WHL draft. It's very expensive ($30k plus per year), and I believe is for-profit.
Yeah, there is definitely a difference in the terminology. Here it seen as Tier 1, which is AAA. These are (supposed to be) the elite players at their age group. Teams can't just decide in the beginning of the season they want to play tier 1. In my district, they have to be approved and it is a process. An examples of tier 1 leagues here is the THF (Tier 1 Hockey Federation) and AYHL (Atlantic Youth Hockey League). THF is owned by a group called Black Bear. Then, there is tier 2. Which is AA and below. Our district has the NJYHL (New Jersey Youth Hockey League) and DVHL (Delaware Valley Hockey League). The NJYHL does have 2 teams in NY. One that has a rink on Staten Island and Bayonne, NJ. The other is Ramapo, NY which is near the border of southern ny and northern NJ. So yes, you could go to a rink in NY and see a NJ banner hanging. In NY, they have a Long Island league. Think is the LIAHA and then the HVL (Hudson Valley League) and many teams will play in both leagues to get extra games and variety. They also have teams from Connecticut. There is also the AHF (also owned by Black Bear) which is for tier 2 teams. However, they are not part of a district at least at 10U and I dont think 12U. Not sure how they work into 14U and older.
If you speak to someone around here and mention the term "club hockey" they are going to think you are talking about any kind of travel hockey, whether it is tier 1 or tier 2. if you say "elite", they are going to think "tier 1". If you say "federation" they are likely just going to think it is just another name for "club" or "travel. Around here, all the tier 1 tryouts are on the same days. The district determines when the earliest day tryouts can happen and then tier 2 is usually about 2 weeks later. Again, district determines the earlies they can start and all teams hold on the same days. Parents then sign contracts after tryouts committing to that team. You can get a release after you sign, but that is between the family and the club and how different organizations handle that varies. I do not know of any geographic restrictions from where teams can draw players in the NJYHL, DVHL, LIAHA, HVL. THF. AYHL, AHF. No idea about when you go "upstate" as you go further north towards Albany or West. as you go towards Syracuse, Rochester, Buffalo. For Tier 2, teams will have a few cuts, but it is not hard to find a team unless you are a novice and just not ready. If for example a team has 25 skaters show up for 12U tryouts, they will likely create 2 teams. If they have 35, they may create 3. In some cases, if a kid does not make the top team for a club, they may not sign with the 2nd if they don't think it will be the level of competition they want. AFTER tryouts, you will see many clubs posting on fb groups that they are looking for a few players to fill out their roster. Yes, some kids will go to more than 1 tryout. And, some teams will have a supplemental or 2nd tryout maybe a week or so later.
If you say "house league" to someone around here, it means at one particular rink. Every rink runs their house league differently. For the most part, it is for the beginner. Skating ability can vary WIDELY. When my daughter played house, she had a year of skating lessons, then 2 years (2 covid years) of the NY Rangers Learn to Play. Some other kids were on the ice for the first time. Some had some skating experience (learn to skate) but first time playing hockey. Our house league has 1 practice and 1 game per week. How it is divided depends on the number of kids, but for the most part it will be 8U and 12U. They will allow some older kids to play 12U house if they are beginners. For the spring, some rinks will have spring house league which is more competitive. Generally set up for their travel players to stay on ice, but without the same commitment. 1-2 sessions per week. Mix of games and practices.