pathfinder-2eshield
I have not found any Shield proficiency in the game. Can Wizards carry around a Steel Shield, and get +2 AC out of it?
What is the point of the Shield cantrip then? Twice the AC, same action. You can cast with a shield in your hand.
Sure, you cannot use Shield Block, but the cantrip allows it only once per combat anyway, after which you lose the AC too.
Light shields prevent you from using that hand for spellcasting.
Each entry there tells you what you can do with that hand when a shield is equipped on that arm/hand:
Buckler: Anything, including casting spells (but with some consequences).
Light shield: Hold items.
Heavy shield: Nothing at all.
Light shields let you hold items in that hand that aren't weapons, and that's it. Spellcasting doesn't qualify.
By strict rules as written, you are correct. The feat is written poorly and has been discussed ad nauseum on the official Paizo forums, with many folks arguing both sides. There are two factors to consider that allow you to clarify the intent of the feat.
First, removing all penalties to hit provides an overly large benefit in comparison to other feats. Most GMs would agree that this is clearly out of line and would not read the feat this way or allow players to use it in this fashion.
Second, (and perhaps more importantly) there is a feat table in the Pathfinder RPG Core Rulebook where this feat was originally printed. I've provided a tiny excerpt of the table below, where the Shield Master feat is listed (pg 116 in the 5th printing). You'll note that the condensed benefit for the feat is listed as "No two-weapon penalties when attacking with a shield". While this benefit description is by no means the be-all-end-all rules source for the feat, it does clarify the intent of the author.
For the two reasons above, it is likely that this feat was only intended to remove the two-weapon penalties when attacking with a shield along with another weapon (options 2 and 5 from your list above).
EDIT: Thanks to SevenSidedDie for bringing to attention the fact that we also now have an official FAQ Answer on this topic which reads:
Shield Master allows a character to ignore the Two-Weapon Fighting penalties on attack rolls with a shield while wielding another weapon, but not any other penalties.
Best Answer
Shields (and their rules) are discussed on page 277 of the Core Rulebook. The Raise a Shield Action is on page 472.
You are correct to note that there is no shield proficiency and that anyone can both carry a shield and use the Raise a Shield Action. This requires buying a shield, occupying one hand with the shield, and spending an action to Raise it. You need to Raise a Shield every round if you want to keep the AC bonus.
The Shield cantrip similarly costs one action. So why bother? For one thing, Shield doesn't occupy your hands. That's worth paying attention to. Most mundane shields cost 1 bulk, which you save with the cantrip. And as you noted you gain Shield Block which you may not otherwise have.
If you do plan on Shield Blocking, remember you will need to budget for repairing or replacing your shield when it's broken. You will also need to continuously spend gold to upgrade your shield, keeping its hardness and HP adequate.
You already noted some of the limitations. Shield gives you a measly +1 AC, which is the same lousy bonus a buckler provides. On the other hand, the increasing hardness is quite nice.
So there is your trade off: a physical shield may offer better numerical benefits, but requires the bulk, gp, and hands to operate. If you are reliably taking attacks and the extra AC or shield blocking are important to you, you want a shield. If you only need that benefit in a pinch or if you need your hands/bulk/gp for something else, consider the cantrip.