Generally I think it's smart for store badges to look like a store badge. My thinking is that users don't read but, instead, recognize the look of the badge. But, I've always modified the standard badge slightly because it needed to sit next to a similar Google Play badge. They are slightly different sizes.

Also, I usually remove the "Download on the" and "Get it on" text, as it is painful to handle translations (hard to image swap to correct language in an email).

However, recently I'm working for a slightly larger brand that is regularly featured on both the App Store and Google Play. After more than 2 years Apple has taken notice of the tweaks that I've made to their badge and asked us to replace it with their supplied asset. It's going to be a lot of work to replace it across all the locations and platforms, and I'm really not sure how to handle the email templates.

Sorry, I don't have any metrics, but I hope this helps.

I think the more publishers use the standard button, the more consumers will view it as a call to action. Kinda like how the save icon is a floppy disk but nobody uses floppies anymore.

I've never thought that badge was a particularly good call to action, personally. It works pretty well as a sign of authenticity though. I usually lean towards a custom one first, because you can make it a stronger on-brand CTA, while still offering visual cues to the original.

If you want to be considered for any sort of feature by Apple, you need to use theirs. Otherwise it doesn't matter a whole lot.