The Library of Congress interprets "a name that is general in nature" in the rule as follows: "general in nature" usually means that the name contains neither very distinctive elements (such as proper nouns or adjectives) nor subject words.

Example A

Otherwise, consider that the name is not "general in nature" and enter it independently with qualifiers as necessary (cf. LCRI 24.4C).

Example B

See also:

24.13. Subordinate and Related Bodies Entered Subordinately