The first sequence was as I envisioned it. Round 2 of the second sequence would be hit twice, the slot increases to 4th level (assuming you can cast a 4th level spell, otherwise you have 1 3rd level and 1 1st level). Round 3: cast fireball as a 3rd level spell and have 1 1st level slot left for next round or cast it as a 4th level spell. Although you could do Round 2: take the attack action, hit once, the slot increases to 3rd level, and cast a cantrip as well. [In general casting a cantrip is a less optimum idea unless you really need a ranged attack, so where for the EK, hit with weapon plus zap with cantrip is the ideal action, here it is not.]
If I was playing this type of PC, the second sequence is what I would go for. The length of combat and your odds of hitting something are the only limitations on how many highest level spells you can cast.* It is a gamble, the traditional spell slot classes know they will be able to cast their highest level spell (at the PC's level) a set number of times per day; the mage knight may not get to cast any at all, but on a day with a lot of rounds of fighting, he/she could cast the highest level spell more than the traditional spell slot classes. For me, I would expect to cast far fewer spells than the wizard (even excluding cantrips), but, on busy days, I would expect to cast more (relatively) high level ones. In some ways it arrives at the same place the sorcerer does when he/she turns low level spells into points so that he/she can cast more relatively high level ones, even though the mechanism is different.
* For the mageknight, the level limiter is more about what you can learn. A 1st level mageknight can no more cast wish than a 1st level wizard, not because he/she lacks a 9th level slot, but because he/she can't grasp the intricacies of the spell until they become a more skilled caster.