Both right, and Passenger's framing is the one we'd use too.
Worth adding how we actually treat it at the moment: the label is the mechanism, not the maths. Every test currently counts the same in the purity average regardless of source - differential weighting isn't implemented yet. We shipped the label first because showing where a sample came from lets you discount it yourself, which seemed more honest than quietly applying a weighting nobody can see. Weighting is planned, it just isn't live.
On how much to discount - what shop-selection actually costs you is that you can't distinguish a shop that sent a representative vial from a shop that sent its best one. Both look identical from outside. It's usually not about fabricated results; a verifiable COA from a real lab is a real result. It's that the sample was chosen by someone with a reason to choose well.
Three things that move it back up:
Volume over time. One shop-provided COA is weak. Twelve across eighteen months, different batches, consistent numbers, is meaningful even though every individual one was shop-selected. Curating one vial is easy. Curating twelve consecutive ones while your actual quality drifts is hard.
Unflattering results. A shop that publishes a COA showing something went wrong is telling you it isn't filtering. We've seen a shop publish a quantity result showing a vial well under its label, then relabel and reprice the product. That single disclosure is worth more than a stack of 99.9% reports, because it demonstrates the absence of selection rather than asserting it.
Quantity data. Shop-provided COAs skew heavily toward purity-only, since purity is the flattering number and cheaper to run. A shop publishing net content is testing something it could have quietly skipped.
On your batch question - you're right, and it's the weakest link in the whole thing. A COA for lot X tells you about lot X. Whether lot X is what shipped to you isn't something the COA can establish.