The cause for confusion is in the implicit assumption of a birth order.
If I have two kids, there are 4 equally likely outcomes:
- boy then boy
- boy then girl
- girl then boy
- girl then girl
Each of which has a 25% chance.
If I give the information "one of my kids is a boy", the only possible outcome that you can rule out is the last one (two girls).
That leaves three equally likely options, only one of which includes a second boy. 1 / 3 = 33%.
If the question was stated with the phrase "my oldest kid is a boy", then you can remove the last two options (GG and GB), which leaves you two equally likely outcomes, only one of which includes a second boy. 1 / 2 = 50%.
The birth order is important, whether you like it or not
The "born on the Tuesday" thing will similarly affect the probabilities.
These concepts are hard to understand intuitively, and you often need to write down all the possibilities before it makes sense (or at least I do!).