IMHO, your second reading is the good one:
The rule is that you cannot check the second parenthesis "(□ or you may search your deck...)" unless you already check the first one "(□ then you may draw a card)".
Which mean, with one check the sentence reads :
"When you would ..., you may discard it instead then you may draw a card."
which is pretty clear
Then with two checks the sentence reads :
"When you would ..., you may discard it instead then you may draw a card or you may search your deck..."
which to me is also clear and reads :
"When you would ..., you may [discard it instead] then you may [draw a card or search your deck...]".
Note that this is consistent with what you would have in extrapolating the then/or priority rule from the "attempt a check" part of the rules:
...“Or” takes priority over “then,” so if a card says “Wisdom 10 or Combat 13 then Combat 15,” you must first attempt either a Wisdom check with a difficulty of 10 or a combat check with a difficulty of 13; after that, you must attempt a combat check with a difficulty of 15.
Which also means
If a card says “Wisdom 10 then Intelligence 13 or Combat 15,” you must first attempt a Wisdom check with a difficulty of 10; after that, you must attempt a Intelligence check with a difficulty of 13 or a combat check with a difficulty of 15.
Thus the application to the Toxicologist power.
The "when you would" must be paired with an "instead". If you interpret it the first way, you get:
"When you would banish a card that has the Alchemical trait for its power you may search your deck for a boon that has the Alchemical trait and put it on top of your deck"
Without the "instead" it's entirely unclear whether you would still banish the card or not, and indeed if you don't banish it where does it go?
Also the last two powers are a much more sensible pair of things to choose between than any other pair.
So it's fairly clear to me the intention at least is as Frencois says, you can't tick the second box until you've ticked the first.