Not exactly. Yes, I want to disable this area.
About buttons, I just want to "My Account" button in TopNav position (float right) works as "log in" button.
Original "My Account" button opens the authorization form on the page (screen1)
screen1.jpg
"Log in" button makes the pop up of the authorization form (screen2)
screen2.jpg
So, I want to "My account" makes the authorization as "log in" does, with pop up.
Hope you get me right

*One notification. When I hover "my account" button,
a link address (left & down of browser page) appears gembiland.com/my-account - the menu alias linked with the module.
When I hover "log in" button, the link address appears gembiland.com/joomla-pages-ii/user-login