There are many different definitions of addiction, so if you're really curious, you might want to specify which particular set of definitions for technical terms like 'dependence' or 'addiction' you're using.

I assume those were years in which he was 11, 12, and 13?

An implied floor(age) hadn't occured to me. I suppose that means that on average the 14 year olds in question would have 0.5 years in which to become a regular smoker. An... interesting... thing to measure. The ones that would be most likely to be smokers would then be those that are closer to 15 than 14.

I meant, first, it's about tobacco and not nicotine, and second, it's a longitudinal correlational study, not causal as your link immediately jumps to (it "creates" a vulnerability).

The sleeper effect held true even if you controlled for gender, ethnicity, socioeconomic status, parents' smoking, and conduct problems

is not nearly enough to claim you have screened off all possible variables and now you are entitled to infer causation. (And the claim is pretty dubious in the first claim: one cigarette does all that? Even stuff like heroin doesn't guarantee addiction after the first injection!)

Going to the full text:

Figure 1​1 shows that 16% (35) of year 7 “one time triers” became current smokers for the first time in year 8 (age 12–13) compared with only 3% (45) of year 7 never smokers. Similarly 18% (28) of year 7 “one time triers” became current smokers for the first time in year 9 (age 13–14) compared with only 7% (111) of year 7 never smokers, and 20% (22) became current smokers in year 10 (age 14–15) compared with 10% (146) of year 7 never smokers. In these respondents, no further smoking, beyond the initial cigarette, had been reported in the intervening years and therefore current cigarette use was not reported until several years after the first cigarette. It was only in year 11 (age 15–16) that new current cigarette use finally equalised across the year 7 “one time triers”, 12% (10) and never smokers, 11% (131).

If there is a causal effect, I like the social suggestion:

Alternatively, from a social cognition perspective,23 an early experience with cigarettes might break down barriers that would otherwise prevent or delay smoking, such as fear of adverse reactions to smoking or insecurities regarding how to smoke. If these potential concerns have been overcome in the past, the likelihood of accepting a cigarette at a later time point may be raised in relation to those who have not had this experience, resulting in the expression of a behaviour which has been dormant. Finally, from a constitutional vulnerability viewpoint, past research suggests that individuals with a particular social and psychological profile are more likely to become smokers.21,22,24 The personal traits that lead to early experience of smoking could contribute an underlying increase in risk of smoking that is not triggered until environmental conditions are right.

Perfectly consistent with a 2:1 odds-ratio, fits with the elimination of the effect by mid-teens, and doesn't attribute implausible powers to tobacco.