I'm not sure if I think there's a difference between depend and rely on my T, but I read your question more in terms of how one's T reacts to your emotional needs vs needs for time or a specific 'thing' taken care of. This is probably because I had a specific need from my T last week to respond in some way to my feeling of fear. I didn't call my T and tried to deal with it myself, which didn't work well. But I wasn't sure my T would either understand the fear I was feeling from our session or that s/he would respond in a way that would make me feel I could rely/depend on them to offer what was needed to alleviate my distress. So, the short answer is, I don't know if I can rely on my T to help me when I'm distressed or in emotional trouble...I haven't tested it.