This is basically the same as for a level-1 heading but I am not sure your reference the correct field.
To get the running chapter name or number in a header,
Document tab. Type is Chapter and Format one of the choices for name only, number only or both number and name.
From the wording in your question, it is not clear if you chose an item in
Cross-references tab, Type Heading which gives you a fixed (frozen, not dynamic) reference to a heading. Those in
Document tab are automatically updated when you cross a page boundary which is not the case with those in
Document tab, you can choose which Level is inserted by selecting it at lower right in the dialog.
You must however be aware of some limitations.
The heading state is captured at top of page for headers and the state of the first paragraph of the text is retained. There is usually no issue for level-1. If the first paragraph is at level-1 and a Heading 2 occurs later in the page, this level-2 is not seen because it was not in effect at top of page. In this case, Writer does not return an empty content but reverts to existing level-1 (more generally if a level does not exist, the preceding existing lower level is returned: 10 → 9 → 8 → … → 2 → 1). A header displays the first level active in the page.
For footers, state is captured at end of page, so you get the last level activated in the page.
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