Thanks. I agree, but some incoming emails from specific contacts, are important and I'd like to see them.
I've tried to convince those contacts to use TEXT MESSAGES.
You could play around with Gmail filters: https://support.google.com/mail/answer/6579?hl=en. I don't think it can modify the notifications, but you could create a filter to automatically star any incoming email from an important contact. Then you could quickly see if you have any emails from those contacts by selecting the Starred label.