Sorry for the confusion, I assume the site in question was built from the older version of X (and Cornerstone) so that is why you have the Classic Section.
Jade's screenshots above are from V2 Sections, If the site is now on the latest version of X and Cornerstone (7.0.4 & 4.0.4 respectively) by default it should use the new V2 Section when you create a new page or add a new section to an existing page.
You can still add a Classic Section to your page while holding down the ctrl or cmd key and clicking the add section in the Cornerstone/Builder, (but we recommend using the V2 Section moving forward).
Please follow the suggestion above of applying ID to your Classic Section.
Hope it helps,