Mei is right to be afraid of love. You can hurt people even while trying to do the opposite. She was trying to close the distance between her and Rinna with a first name only approach but it only showed Rinna that their feelings towards each other is very different. Rinna is in love and she is naturally very apprehensive about their relationship.
Mei isnt in love like she is and thus very non-chalant about such an arguably important step.
Human communication is hell.