I am so glad to hear I am not the only one with this problem. I registered just to post on this.
I want Gmail notifications to never vibrate. Ever. When the ringer volume is up, it will work properly by using the only ringtone I have assigned it and nothing else.
However, the malfunction occurs when my phone goes into vibrate mode when I'm in the office. Despite the proper settings, my phone will vibrate when I get a new email.
Settings >
user@gmail.com > Notifications (checked) > Inbox sound & vibrate
> Label notifications (checked)
> Vibrate (unchecked)
> Notify for every message (checked)
Choosing a ringtone to none doesn't solve the problem because then I won't be able to get audible notifications when I do want them. This must be a bug with the S4, my previous HTC phones never had this unsolvable conflict.