Got a tick fully embedded?
Use a tweezers.
Think you need oil?
Use a tweezers.
Ready to try a cigarette, or lit match?
Use a tweezers.
Got one in a bad place?
Use a tweezers.
Can't reach it?
Have someone else use the tweezers (they may need instruction, depending on locale...)
If all else fails, head to the doctors office (where they'll use tweezers...).
I "live" in the outdoors, and ticks are very common (both wood ticks and deer ticks). A week ago I was in the woods for an hour, and found eight ticks on me (none embedded yet). The most I've found on me was 11 in one day. My dogs constantly bring in ticks. So do my kids. And my wife. We find ticks almost daily in the spring and summer.
The OP posted good info- the CDC has it right... The longer the tick is embedded, the more likely you are to have disease transmitted. Oil, or the often-substituted Vaseline is way too slow, and often the tick simply dies (while still embedded) before it releases. Cigarettes (or burning matches) add a burn possiblity, which increases the risk of infection at the site of the bite. And often results in a dead (burnt, and still embedded) tick.
My brother in law now swears by the tweezers too. He had a tick embedded under his "family jewels", and enlisted his wife to aid him. She first tried dousing it with fingernail polish remover. When that didn't work, she decided to try a lit match. Flammable liquid/family jewels/fire= BAD! In desperation he called the doctor, who recommended tweezers.