Looks to me like you can designate two per year as June 1 cuts without waiting until then.
I think they've changed this several times in the cap era.
Yes you can designate two as June 1 cuts BEFORE June 1.
But you can't use the cap savings until that date. The benefit of designated someone a June 1 cut before June 1 is that it allows you to spread the cap charge over two years but it also allows you to avoid paying more bonuses, etc.
To put numbers on this. Let's say the Cowboys want to cut Player A. When they cut him, they have to eat $20 million in cap charges. If they cut him before June 1, they advance the $20 million against this cap. If they cut him after June 1, they can split the $20 million between 2022 and 2023.
However Player A has a $4 million roster bonus due the 10th day of the league year (let's say that date is Feb 27th). If the Cowboys cut him Feb 26th and designate him a June 1 cut, they only have to take $20 million in cap hits. But if they wait until actually June 1 to cut him, their cap charges are now $24 million.