Yes, there is a correct way to sing without belting or projecting.
Singing from the diaphragm will absolutely take your singing to the next level - it just takes discipline to develop. Hang in there. Singing technique is no joke and WILL make all the difference in your singing career, be it for personal enjoyment or fame. However, if you feel you aren't being given proper instruction seek a lesson or two from another teacher.
Some other things to consider:
Volume < Tone
Being able to keep pitch is way more important than keeping volume. While the two sometimes go hand-in-hand, I cannot stress this enough. Your voice is precious, no matter how long you have been singing. Belting has it's place, in music, but seek to maintain pitch above all - I am sure you do with your vocal teacher. Diaphragm technique takes time to develop. Eventually, you won't even think about it anymore.
Plenty of singers have volume issues, do not fret! Singing loudly isn't always sought after. What matters is pitch. Pitch, pitch, pitch.
Few are born with the natural ability to project their voice. The majority learn from a teacher - even those with natural talent need guidance. Check out my response to a similar question here.
Finally, SE is an inadequate place to learn singing technique. I do not mean to discourage your question - in fact, I encourage these questions! However, the truth is you cannot learn proper technique from personal trial and error. Singers need guidance as singing involves your whole body - its quite aerobic.