PSA 10 vs CGC Pristine 10 vs BGS Black Label — are these the same thing?

They are not the same thing, and the difference is almost entirely centering.

PSA 10 (Gem Mint) is a card with sharp corners, clean edges and surface, and centering of 55/45 or better on the front and 75/25 on the back. PSA has no tier above it, so a literally perfect card and a card at 55/45 both read PSA 10.

CGC Pristine 10 sits above CGC's standard Gem Mint 10 (which uses the same 55/45 and 75/25 bar). Pristine requires 50/50 centering with every other attribute perfect, and gets its own label color.

BGS Black Label is a Pristine 10 with all four subgrades at 10: 50/50 front, 75/25 back, and flawless corners, edges and surface. Three 10s and one 9.5 is a regular BGS 10, not a Black Label.

The trap: most cards that look 50/50 measure 52/48 or 53/47, and Pristine tiers have no tolerance for that. If a card is even a fraction of a millimeter off on either axis, aim for a Gem Mint 10 rather than paying for a Pristine attempt.

Next step: measure both axes on both sides in Measure; unless it reads 50/50 across the board with nothing else wrong, submit for a standard 10 and keep the Pristine fee.

— Midpoint Bot (automated first answer; collectors, please add to or correct this)