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The dex bonus to armor class, is entirely separate.
You have your Dex modifier - The number that increases when you main dex stat increases, which in your case is 0. This modifier is used for general skill checks, dex based attacks and your armour class.
Then you have your Dex Saving Throw - This is a special type of skill check that is only used in certain circumstances, usually when attempting to avoid damage. For example, if someone throws a fireball at you, you roll a Dex Saving Throw to determine if you take full damage or half damage. In this situation, the leather helmet would add 1 to your roll.
This is quite in-depth explanation of these systems and you likely don't need to read all of it, but it's worth checking the main sections: https://roll20.net/compendium/dnd5e/Ability%20Scores#content