Below is the excerpt from English Proficiency Test
Our professional designers produce high-quality custom logo designs (a) that exhibit just the image (b) that our clients want to project.
Question 1 Are there any difference between (a) that or (b) that grammatically?
If so, what would be the proper term for denoting their difference?
Question 2 In informal style, one can leave out the relative pronoun. However, as far as I know, this leave-out permitted only in case of in that relative clauses, the relative pronoun denotes the intention/object of the verb.
In addition to it, we can’t leave out the relative pronoun when it is the subject of the verb in infinite relative clauses. Do I understand correctly?