Good thinking, but not quite right. The three steps of spellcasting are:
1. Cast spell
2. Counter spell
3. Resolve spell
All spells pass through this process when they are played. Then, depending on type, some more things may happen. An attack spell, for example, then goes through the 8 steps of an attack.
Enchantments are special in that they resolve twice. The first time as a cast face down enchantment and the second time as a revealed enchantment.
This means that the Gate triggers when the spell is Cast and Resolved the first time as a face down spell, but doesn't a second time, even though it is resolved again, because it was only cast the once.
If you'd like to know more about this, check the rules v.2 pages 13, 14, and 18.