Hi Zeljka,
There are a few ways I know of you that might like to try.
The simplest option is to simply move the inner rectangle manually by selecting the text box. You may have already tried this.
If you use the format painter from an object where this is already applied, the same positioning for the text box should appear on the other symbols.
You can also control the positioning through the template that you are using. Go to Administration > Configuration > Conventions > Templates and choose the template you are applying to the model. Edit this template and go to step 4 in the wizard. Click on the 'interface' symbol and then the 'Name' attribute. You can then define the default position for the attribute (as well as many other settings).
Finally, the default position of the name attribute is aligned to the symbol's "inner rectangle". When you design or customise a symbol in the Symbol Editor, you can draw a rectangle and right-click on it to "use as the inner rectangle". So a potential solution (but probably a last resort) would be to define a custom symbol with a new inner rectangle.