I'm sure she wasn't criticizing you - she was probably complimenting you on raising a cooperative child. Your child obviously has learned about boundaries. Or, she was stating that all kids tend to behave better for someone other than a parent. We all go through that, no matter how many kids we have or whether or not we have a spouse/partner.
Don't be so hard on yourself. Kids are usually tougher with their parents because it's a power play and because they are more comfortable/relaxed and they know you love them unconditionally. So they aren't risking losing your love by being pains in the butt. That's good, but frustrating, you know? They also worry about impressing a sitter. Same reason kids behave in school - there is peer pressure, a "stranger" (the teacher, the principal), and also there are immediate consequences (loss of privileges) rather than just parental annoyance. Teachers can also send disobedient kids to the office - parent have to be more creative.
So, get through the basics of the bath as quickly as possible - the cleaning & shampooing part. If your child cooperates, then there's play time afterwards. If not, in and out in 3 minutes. Don't argue or plead - just be in control and business-like.
Don't apologize for needing to go out. You are an adult, and all of your needs cannot be met by a child, no matter how much you love that child. See if you can trade off some babysitting for free now and then - either with one person, or by forming a co-op with a group of single parents. Your child will do better if he sees that you are a whole person with needs and wants, and that you are not a slave to his schedule and desires. He will learn to treat women properly if you show him what women deserve - which is to be a whole person and not just a mommy.