Your screen shots both shows a drop-down on the QAT (rather than the Ribbon). I do not believe that what you are using in Word 2007 is exactly available in Word 2013 but you should be able to have something that makes you happy. Note that the display of styles in the Styles Pane can be set to show what the style looks like as well. You do that by checking the box for preview at the bottom of the Styles Pane.

Here is the QAT modification dialog showing three choices.
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Here are screenshots of what you get.
First is the Styles Gallery Classic dropdown and closest to Word 2007. Note the little window that goes on the QAT. You can type the name of a style or its shortcut in here as well. This is Choice 1 in the screenshot above.
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Here are the Ribbon Styles Gallery (choice 2) and the Styles Pane (choice 3). The classic Styles Pane is occluded by the Gallery. It is my preference.

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I actually use both the classic drop-down (Choice 1) which includes a window showing the current style with the button for the Styles Pane (Choice 3). The Styles Pane can also be toggled with Ctrl+Alt+Shift+S. (Yes, being a piano player would help.) In either choice 1 or 3, you can use the Manage Styles dialog (button on the Styles Pane) to organize which styles are displayed and the order of display.

Please see this article for more on Modifying the QAT in Microsoft Word. The default storage location for such modifications is not the best one.