Remember that the internet is full of smiley young people who've done a two-week teaching course and think that they're experts. Lucy is very pretty and charming, but she's far from an expert. She's taken one rule - that you can't use 'that' in a non-defining clause - and extended this idea to invent her own rule that you therefore must use 'that' in a defining clause. This simply isn't true. By the way, some of her other explanations are also doubtful, especially the first who/whom question.

It would be clearer if you said 'the books which you bought for her' or 'the books which you brought to her.'

You're right. You can use 'which' or 'that' in a defining relative clause.