That might be psychological, to some extent. Just like most people need to pee when they hear or even imagine running water… your brain has linked up certain cues. Since you’re aware of this phenomenon (as proven by your question), you’re looking out for any signs of needing the bathroom as you drink. At other times, like when you’re out or busy, you might not be so attentive. So you’d miss the times that you don’t need to pee right away.
Does this happen always, or just sometimes? Because I recently learned that when we drink while our bodies are dehydrated, the water gets absorbed by our mucous membranes and never makes it to the bladder. So that might explain why only the third glass of water actually creates pressure on your bladder.