You make a very good point. Serial Killers do like to define themselves by choosing to claim or highlight certain murders, while not admitting to others. A perfect example would be killers who do not want to admit to killing a child and even get very upset when someone suggests that they are guilty of it. A homosexual who kills women might not want it known that he also kills men.
A basic and common trait of a serial killer is that they have absolutely no feeling for their victims. Whether they actually like killing people, or simply feel compelled to kill is debatable, but the fact is that they do it without any sympathy for their victims. Perhaps they simply start with killing anybody and later focus their efforts on more specific victims in their crazed reasoning that what they do is for some higher "cause" or "quest".
The problem with comparing a well known/defined serial offender is that often such an offender is ruled out as a suspect based on the idea that their "usual MO" doesn't fit a given crime.
Ted Kaczynski openly admits to the bombing attacks tied to him, and states that resulting injuries and deaths were a necessary part of his "war against technology". But when did he first take on this role? Had he committed other bombings earlier? Had he killed earlier?
The same could be asked regarding other serial killers. On one hand, just because a serial killer was in the vicinity of a murder, does not mean that he did it. Some folks try to find connections for almost any crime to a given serial offender. But on the other hand, he shouldn't be eliminated out of hand simply because the murder does not fit all of his earlier (or later) killings.