Hello,

I had the same situation with our company in v14 (same case in 15). After doing a bit of research and playing a bit with the system, I found out the following:

The setting that takes care of this is called a subtype in Odoo and it can be found by going into debug mode and under Settings > Technical > Subtypes. 

Search for the message type that is called "Note." Make sure that the Internal Only field is enabled. When this field is enabled, it doesn't matter that the button is visible or employees only, the log notes will not show for the customer. If the field is not enabled, then you need to manually make the button visible or employees only.

I made a short video demonstrating this behavior in a runbot.

My only issue is that if the button shows visible, it creates panic for the employees thinking that customers can see the internal discussion of the company, which happened to me. I'll update once I find a way to make the button employees only by default.