As of December 2018, only Shopify Plus shop-owners are able to disable Shopify's own order notification emails. You can supplement these with Mechanic emails for confirmation and/or fulfillment, but they can only be disabled for Plus accounts.

If this is important to you, and you're not a Plus account-holder, send an email to support@shopify.com and let the folks there know that this is affecting your business.

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