In T1, when Sarah Connor crushes the first T800 in the hydraulic press, both cells should have been damaged, and there should have been a catastrophic failure. This obviously did not happen so we know that this version of the T800 did not come with the same power supply as we see in T3. Because Judgement Day was not postponed by this event, technology available to SkyNet did not improve for T2 in the T800.
At the end of T2, Judgement day is postponed, and humanity continues to innovate. New technology is available for when SkyNet become self aware in T3. This new technology included much better processing power in modern computer systems, as well as an exponential increase in processing nodes on the internet between the original 1997 J-Day and the 2004 J-day. SkyNet had a better start in the altered future, and could therefore dedicate more resources to developing better technology earlier on in its existence, possibly altering the design of the T800 to include previously unavailable technologies.
In T3, when the Terminator was explaining the damage to John Connor, he explains that he is powered by two hydrogen fuel cells. Hydrogen fuel cells are non nuclear, but instead rely on chemical reactions to generate immense heat for power. This is evidenced in the movies by the fuel cells glowing a molten red color.
In T:Salvation, John Connor explains that the Terminator fuel cells are nuclear. This is an easy mistake to make as many people who saw T3 witnessed the explosion and thought "Hey, that was a tiny hydrogen bomb!" John Connor, as we know, is not a nuclear physicist and probably would not be able to explain the differences between a hydrogen bomb and a hydrogen fuel cell.