Singer for over a decade here. Belting out your favorite songs does count as a form of practice to a point. The vocal exercises are important to get your voice into optimal shape and more importantly especially for beginners, nail down technique. It's vital if you want to be a singer where you're going to be performing many shows or recording long term. If you're singing incorrectly and putting strain on your vocal chords, even singing clean, over time you will damage your voice. Damaged vocal chords, callouses, nodes (that require surgery to fix and your voice will still never be the same).
If your vocal coach is doing nothing but showing you the boring, annoying exercises, see a better vocal coach who focuses less on exercises and more on letting you sing the songs YOU want to sing and addressing any bad habits and adopting proper technique where necessary. You might need a complete overhaul or just a little fine-tuning. Seeing a great, classically trained vocal coach massively opened my range to where high notes I was straining a bit before, I can reach easily and beyond by learning the proper techniques. Don't be too put off by having to make some embarrassing sounds before you get the hang of it. It took a bit to get used to but it was one of the best decisions I've made for my voice as I have been singing, crooning, belting, growling and screaming (which is also very important to use proper technique or it's a fastrack to a destroyed voice) all in the same performances with no problem at all.
After reading through the comments I should add I had been singing a while before and already had the tone and range to start with, I have no idea if one can be taught to sing if they don't already have good pitch, rhythm and tone. My girlfriend is a singer too, she never had lessons but has so much raw talent with pitch and rhythm (I had no idea she wasn't quantising her keys and didn't even know what it was, she was just that on time.) She learned just by copying her favourite singers and realizing she could do it. I've just given her a few pointers to make sure she isn't doing any damage. Some people are just freaks like her. I have perfect pitch (which means if you play me any note I can tell you what it is, tell you all the notes in a chord and produce any note you ask me without a tone reference) It took me a lot of practicing to be as precise rhythmically as she just is from the womb!
Listening to your clip, you're in a good place to start. You've got a nice timbre, you're obviously not tone deaf and understand pitch. I think you all you have to do is keep singing, keep recording yourself and you'll get more confident and better each time.
