Specific equipment can make for longer draws or more consistent draws, but you should be able to draw with just about anything - on any cloth.
My problem was drawing too far or too short depending on the condition of my tip. i.e. inconsistent draw. So I worked on this for two months. I went with a dime shaped Moori III Q (hard) pig skin tip. I keep my tip dime shaped and resurface it about once a week with a sandpaper dime shaped shaper. I just lightly sand "one time around" so the tip has a new surface all over. I need a new tip about every 6 months, but I can replace my own, so no problem.
But you should be able to draw with any tip or any tip shape. I feel that keeping the tip a specific shape and always the same surface condition will eliminate a lot of variables - and help you to learn to draw specific distances - each time, every time.
So the other night I'm playing and need to draw back 6 inches. I did it. Next I needed to draw back 1 ft. I did it. Everyone is saying "nice draw". Well I have worked and worked on this, so no big deal for me.
The first thing is to learn to draw. Follow through until your cue tip stops 6 inches past where the cue ball is. Have someone watch you. Slowly stroke the cue ball - just enough speed to make the ball into the pocket - then leave the tip of your cue STOPPED 6 inches past where the cue ball was - touching the cloth on the table. Leave it there for 4 seconds!
Have firend make sure you are doing this every time.
Stroke slowly - just enough speed to make ball into pocket.
Chalk tip well, especially around the sides of the tip.
Hit low on the ball, follow through, hit slowly.
Practice - Line up row of balls across table. Place cue ball 1 diamond back from object ball. Shoot with draw to make ball into corner and draw back cue ball 1 diamond. Keep practicing (every day for 2 months).
Note: The cue ball spins backwards when you hit it with draw. As it rolls down the table, the backwards spin stops, then the ball slides, then the ball begins a forward roll. Thus draw shots where the cue ball is a long distance from the object ball are very difficult. Practice with a striped ball as the cue ball and watch what happens as it travels down table. May want to video tape it to see it in slow motion.
High speed video "close-up" of a good draw shot...
http://www.engr.colostate.edu/~dga/pool/high_speed_videos/new/HSVA-38.htm