你说的情况在不少语法书确实有点“乱”，帮你找到一个权威的说法。以下材料引自著名语法学家夸克等人合著的《英语语法大全》：

There is, however, one circumstance in which the zeroarticle occurs in such constructions: this is wherethe complement (or an equivalent appositive noun phrase) namesa UNIQUE ROLE or task. As the following examples show, in such cases thezero article alternates with the:

但在如下的情况下，这些结构中出现零冠词，即句子的补语（或相应的同位语名词短语）是某一个独一无二的身份或职务的名称。下列例句表明，在这种情况下零冠词与the可以交替使用：

Maureen is (the) captain of the team.

John F. Kennedy was (the) President of the United States in 1961.

As (the) chairman of the committee, I declare this meeting closed.

They’ve appointed Fred (the) treasurer, and no doubt he will soon become (the) secretary.

Anne Martin, (the) star of the TV series and (the) author of a well-known book on international cuisine, has resigned from her post on the Consumer Council.

根据夸克等人的《英语语法大全》，你说的这种情况中的冠词是可有可无的（并不是一定要省略）。