With "being" vs without "being"

Here are two sentences:

1. The training examples get labels 1 or 0 based on the tactics being useful or not for the proof state.
2. We take the inspiration from the paper and implement an online version capable of incremental optimization.

My English teacher told me that for the first sentence, either "tactics being useful or not" or "tactics which are useful or not" is fine. However, "tactics useful or not" is awkward in her opinion.

For the second one, she said we could use "an online version capable of incremental optimization" instead of "an online version which is capable of incremental optimization".

Why in the first case, we cannot ignore "which is" or "being"? I cannot understand why after we removing "being", the sentence "The training examples get labels 1 or 0 based on the tactics useful or not for the proof state." sounds awkward? For me, "on the tactics useful or not" is just an abbreviation of "on the tactics which are useful or not". I think the meaning of "on the tactics useful or not" = "on the tactics which are useful or not" = "on the tactics being useful or not"

You can, in the first example, omit 'which is' or 'being'. You can say, "The training examples get labels 1 or 0 based on the tactics, useful or not, for the proof state."

The reason you can say this is because 'useful or not' is an adjective phrase modifying the noun 'tactics'. And adjectives can follow the nouns they modify. These are called postpositive adjectives.

Please consult the linked websites for further reference. Also note that in this case you would need to bracket the adjective phrase with a couple commas as I did.