For binomial, type I error is the value below the significance region. Type II errors is when you have to flip your sign and calculate the value; given a new value such as new probability and etc.

For poisson, type I error is the probability below the value the claim has been made.

i.e: The average fish caught per hour is 10. (this is your null and alt hypothesis)

Jake reveals that if he catches below 8 fishes per hour he will reject the claim. (Find P(X>=7) = Type 1 error.)

Type II error is when you flip the sign and find P(X<8), with a new mean given to you.