I think there is little difference between 犯罪者のこと and 犯罪者. I think user54609's explanation is nice.
罪人：犯罪者のこと basically means
犯罪者のことを罪人という. You explain
罪人 by rephrasing it saying
つまり～のこと is a very frequent expression used to rephrase what just has been said.
AはBのことだ basically means
A means B or
A refers to B. You can't expect
A is a kind of
こと because some nouns work like verbs by connecting two phrases in some way. e.g.
AはBの感じだ, etc. When you put them into English, they become
A plans B,
A means B,
A seems B, etc. And
AはBのことだ is similar to them. It not
A is a koto of B, but
A kotos B. i.e.
A refers to B
こと might originally mean something like 言うこと/ことば or 考えること. It has become an abstract marker for objects related to mental activities. e.g. 友人のことをどう思う, 友人のことをどう扱う, 友人のことをどう考える, 友人のことをどう呼ぶ etc.
I think のこと should be redundant because the verb is already a mental activity and require a mental object. But it's just required for some verbs and can be omitted for some else verbs, just like のところ is required for some verbs, even if the verb obviously requires a locative object.
(人)のことを言う and 話す are more likely to be used with ～のこと, but in ～のことが好き and ～のことを指す, こと can be omitted. I don't think they make big difference.
Maybe it's off-topic, but I think I sometimes saw こと used for non-mental-action verbs. e.g.
I don't know what the nuance is, when and how it is used. But I guess it might be similar to 扱う.