Sure. Looking at my post above, I should have made clear that "proven record of -ing" is not the only idiom that is connected to the word record. There are other idioms connected to other meanings of the word 'record'.
However, looking at your examples, we can see that we're not dealing with a physical vinyl record (your third example), or a specific piece of evidence (your first and second examples).
If I want to say that Alina has a reputation for being late, the idiom is 'record of -ing', i.e. 'Alina has a record of being late.' The sentence 'Alina has a record that is late.' doesn't convey my meaning.
Of course, you can find the words 'record that' together, but this would probably be a 'NOUN that' modifier. For example, using the example above, I could give some more information about Alina's reputation by saying 'Alina has a record of being late, a record that everyone is aware of.' I hope you can see how the modifiers 'of being late' and 'that everyone is aware of' give different kinds of information about Alina's record.
I recommend using a good dictionary to check the meaning of unfamiliar words, such as
https://www.lexico.com/en/definition/record