From the CDC:
How to remove a tick
Use fine-tipped tweezers to grasp the tick as close to the skin's surface as possible.
Pull upward with steady, even pressure. Don't twist or jerk the tick; this can cause the mouth-parts to break off and remain in the skin. If this happens, remove the mouth-parts with tweezers. If you are unable to remove the mouth easily with clean tweezers, leave it alone and let the skin heal.
After removing the tick, thoroughly clean the bite area and your hands with rubbing alcohol, an iodine scrub, or soap and water.
Avoid folklore remedies such as "painting" the tick with nail polish or petroleum jelly, or using heat to make the tick detach from the skin. Your goal is to remove the tick as quickly as possible--not waiting for it to detach.
CDC - Tick Removal - Ticks
If a person removes a tick by any other method it may get the tick removed.
But the above is the best way.
"Tick vomit"? Well, the technical word for that is regurgitation-
--- but it doesn't matter.
Why the CDC recommends grabbing the tick as close to your body as possible is to try to prevent the tick from regurgitating the contents of it's midgut (which would be squeezed in the process) which is where any of the bacteria that cause Lyme will be. IOW, you don't want to angrify the tick.
And a doctor who specializes in Lyme told me --- never ever remove a tick (from an animal or yourself) with your bare hands!