I've been using the ADDICT system, but after reading this thread I had another look at the combat chapter and I think I've figured it out!
The rule on p 65 is used when the combatant attacking the caster does not have a weapon. The rule on p 67 is used instead when they do.
Spellcasters do NOT simply begin casting at the beginning of the round, and then the spells goes off at the end of the casting time. The time at which the caster begins casting depends on the initiative dice, and then completes after the casting time in segments. This seems very clear looking at p 65: "Their commencement [of spell casting in melee] is dictated by initiative determination as with other attack forms, but their culmination is subject to the stated casting time. Both commencement and/or completion can occur simultaneously with missile discharge..."
The rule on p 65 for attacks without weapons says "Attacks directed at spell casters will come on that segment of the round shown on the opponent's or on their own side's initiative die, whichever is applicable" because the attacker may have to wait for the caster to start casting before interrupting them. Here's where I extrapolate a bit but...the caster begins casting on the segment of their opponent's initiative die roll, and finishes after an additional number of segments equal to the casting time of the spell. If the attacker loses initiative, their attack comes on the segment equal to the caster's die roll (the higher die roll), and they therefore have a chance to interrupt if the difference between the initiative dice is less than the casting time of the spell (ie if the caster's initiative die < attacker's initiative die + casting time). If the attacker wins initiative, they go on the segment of their own die roll (because they're waiting for the caster to begin casting) and they always get a chance to interrupt (because the caster will take at least one more segment to complete the spell).
The rule on p 66-67 assumes that attackers with weapons also always get a chance to interrupt if they win initiative and the rule is meant to give them a chance when they lose it (ie the losing die roll will always be the attacker's). This isn't stated clearly anywhere but I'm going with it based on the fact that the example given assumes the attacker lost initiative and twice refers to the attacker as "reacting" in this situation. This rule does NOT use the initiative dice to determine at what segment the attack comes (it just says "the blow...[arrives] either as the spell is begun or during the first segment of its casting". Using the unmodified speed factor when initiative is tied is weird, but slightly less so when you consider the symmetry with the procedure for deciding ties between two weapon-using combatants (if both combatants have weapons, SF vs. SF; if one is a caster, SF vs. casting time).
Putting things together:
Spell casters complete a spell after a number of segments equal to their opponent's initiative roll plus the spell's casting time.
Attackers have a chance to disrupt if they win initiative (the attack comes when the caster just begins casting, therefore on the segment equal to their own initiative roll).
If the attacker loses initiative, they still have a chance to disrupt. If they do not have a weapon their attack comes at the segment equal to the caster's initiative roll, and can disrupt if the caster is still casting at this time (ie if the casting time of the spell plus the attacker's initiative roll is greater than the caster's initiative roll). If the attacker has a weapon, then they have a chance to disrupt if the difference between their losing die roll and their weapon's speed factor is less than the spell's casting time. If so, the attack is assumed to arrive just after the caster begins casting.
Missile weapons use the p 65 procedure, but modified by the attacker's Dex Reaction/Initiative mod. So a missile firer with really good Dexterity (and unencumbered, see PHB 101) is the best way to disrupt spell casters in practice.
