How to parse this phrase?
The former interpretation is correct.
Chinese characters are composed of components (some radicals, some phonetics, some neither). For many of the components, there are standard or widely used ways to refer to them. A common pattern is {component}+字+{旁/头/盖/底/儿/框}. (The last part refers to which part of the character it is--e.g., side, top, bottom, &quot;frame&quot;) For instance:

讠= 言字旁 e.g, 说、讲
⺮ = 竹字头 e.g., 管、篮
心 = 心字底 e.g., 忌、忍

Some of them break the pattern a bit:

刂 = 立刀旁; e.g., 刻、则
彳 = 双人旁; e.g., 很、彼
忄= 竖心旁; e.g., 快、慢
辶 = 走之底; e.g., 这、迅

Sometimes the situation is complicated because the component does not exist as a standalone character. In that case, it is often named by a common character with that component

宀 = 宝字盖/宝盖头; e.g., 宇、定
 = 春字头; e.g., 泰、秦
冂 = 同字框; e.g., 周、网

Keep in mind that there may be multiple names for a given component.
Here are some resources to look at.