Does Babyhead’s Tighten the Bolts ability trigger if I use a payment modifier to play a higher-cost card for 2 {I} or less?
Yes. That ability only cares about the total ink cost paid to play a card, which is determined after all payment modifiers are applied. As long as you paid no more than 2 {I} to play a card, the Tighten the Bolts ability triggers. For example, if you played a cost 3 card and modified the payment to pay 1 {I} less, you paid a total ink cost of 2 {I} and the ability triggers.
Official set release notes — Wilds Unknown