As far as I know, there is no rule covering this. That's because there isn't a restriction on whether or not you can use "that' or "to+infinitive", but rather a matter of in which grammatical situations you can use those constructions. For example, when I looked at your examples, it immediately occurred to me that there are grammatically correct constructions you can make with BOTH "that" or "to+infinitive" depending on what you are trying to say. Does this help, or is there something else you're trying to ask?

Thanks Danielle.
If I remember correctly (i dont have the OG at work ), the explanation for one of the question said that we need 'that' after the verb. I might have misread something, so I need
to recheck.

Check out this link which mentions something on 'that', 'to+infinitive'.
https://www.edufind.com/english/grammar/rep7.cfm

B and N, I checked out the link, and those tables, while really helpful, are dealing with idiomatic/formulaic constructions. They are not grammatical rules of construction per se. Those tables show the way things are LIKELY to be constructed IN GENERAL. And, as we know well, just when you think you're getting the hang of it, you get a question testing some part of grammar that is relatively obscure. Frustrating!

->you get a question testing some part of grammar that is relatively obscure. Frustrating!

exactly. thanks a lot for your help. just out of curiousity, what has been you max. score on the verbal (my guess is b/w 49-51).