Is a new iPhone from apple.com paid via Verizon or T-Mobile going to be locked or unlocked?

If you purchase an iPhone at full price directly from Apple and without a carrier subsidy, then the iPhone will arrive unlocked and pre-loaded with your carrier plan. If you opt for a carrier subsidy where those are offered, you’ll usually get a carrier-locked iPhone. If you purchase an iPhone via a carrier, you’ll usually get both a subsidy and a carrier-locked iPhone, unless the carrier specifically indicates otherwise.

See Note in the screen shot you posted.

Only iPhones bought with AT&T's installment plan will be locked. Choosing a carrier does not mean it will be locked to them. Just that it may include a SIM card for that carrier and can let you setup your service with that carrier directly drink the purchase process. If bought with Apple's now Installment plan, it will not be locked.

Apple sells unlocked phones. Not sure what you are asking when you say "paid via verizon or t-mobile".

If you specify that you are going to use a given carrier, Apple may provide a phone that is locked to the carrier that you specify, especially if there is a special promotion in effect at the time.

Obviously for example, if you go to a Verizon store to buy an iPhone, the phone is going to be locked to Verizon's service.

Your carrier plan can be pre-loaded. With a full-price purchase and no carrier subsidies, only pre-loaded. Not carrier-locked.