Not That Coordinated

[NovaUltimatum]

Which job has less casting to do while actively dodging, Black Mage or Summoner?

Absolutely SMN.

While you can do all content in the game as a BLM, the only way you can keep up with the other DPS roles in a good group is to follow the caster rule of ABC (Always Be Casting). As BLMs inherently have the longest casting times, they're really not built well for fights that require frequent moving, unless you are experienced enough to predict all of the fights mechanics ahead of time so you can cast in a rhythm that doesn't interrupt your casting.

SMN on the other hand, has a pet that for most fights can remain stationary and will constantly be contributing damage for you. Couple that with the fact that SMN is built upon doing huge amounts of damage from DoT effects, where once the boss has been DoT'd, it will just keep ticking away damage while you're moving. SMN also comes with 4 nice ways to add damage, while you're dodging as well (These are Ruin2, Fester, Energy Drain and Swiftcast->Shadowflare) All of those abilities are on cool down so can be guaranteed to work for you if you just use one that's available.

Attacking while moving on BLM is also possible (Scathe, Thundercloud Proc, Firestarter Proc or Swiftcast->Flare), however the good abilities are proc dependant so require a bit of luck (while you can just not use your procs until you have to move, most BLMs prefer to use a proc as soon as it's up so another one can come). So essentially the only available attack guaranteed for you while moving is scathe, which is gonna have pathetic damage.

Boosting your Spell Speed can help a lot with that. I found it very hard to cast when dodging a lot with low SS.

+1 to this. If you're adamant about going BLM then it will help you to build Spell Speed more than Det/Crit.