STAHS uses the Cambridge Latin Course to teach Latin. I'm not sure about Habs - does anyone know? This course is half history/half language. It's been taught in many schools for many years. It's appeal is that it 'reaches the masses' in that it maintains the interest of all children, regardless of ability, as it covers lots of historical topics as well as the language itself. It is rather different from some Latin courses of 20 years ago. The CLC can get rather slow and dull for bright children. I would ask the schools to confirm which course they teach.
Either way, if your DD is already keen on Latin, that will continue if the teachers are good and inspire her. Have you met them and asked how they keep their pupils interested in the subject?
There are so many others important matters to consider when trying to decide between schools, especially as far as girls are concerned - what is the extent of mental health issues? (HUGE issue in girls' schools at the moment), what is the mobile phone/access to social media during the school day/social media policy? (this can have a HUGE impact on learning and a girl's development). These are things which you tend not to know about before choosing a school and yet are crucially important. Some indie schools require girls to hand in their phones at reception on arrival, whereas others, including STAHS allow them to be used in lessons (which of course results in abuse of the system). Also, think about whether you prefer a city centre school or not - that makes a difference too!
Habs uses the Cambridge Latin course too. My DD is really enjoying the Latin at Habs and the teacher is very good.
Re mobile phones. The girls have to turn off their phones and keep them in their lockers throughout the school day including breaks and lunch. If they need to make calls they have to go to the school office. The school is very hot on misuse of the Internet and social media and we get regular communications from them when they have spoken to the girls about the issues. They give very good guidance to new parents about their expectations of Internet/social media use by the girls.