I don't think it's chasing after someone. I tend to do the same thing. Though to be honest calling a client just ten minutes into a session when they don't show up seems a little soon to me. I'm sorry to hear that you felt bothered and intruded upon, but at least in my country this would seem to be normal. Your therapist needs to know if you're not going to show up just as any other professional would (or really any other person). She could just spend that time doing something else or having an appointment with another client. And yes, given the nature of psychotherapy and perhaps other circumstances, there's the added element of working with vulnerable people who may not show up due to difficulties which would be better addressed, and in my opinion it is the therapist's job to show availability (definitely not chase after clients and insist though).

If this is bothering you, it's probably best to let your therapist know and negotiate boundaries at your next session. I hope your vacation has been a good one anyway.