No, they didn't. Final Fantasy XI had WHM and that was it. RDM was forced into healing (and haste/refresh spam) and SMN was shoehorned into healing. That's very different from having multiple healing classes.
That being said, I don't understand where this allergy to jobs being interchangeable comes from. Something has to give in order to avoid the crap from XI where you had the "ace" jobs that were used for events and everyone else was just fodder. Not to mention in order to kill off what we're currently seeing right now where people go for certain set ups and just lol at others. What some don't seem to realize is that asking for niche jobs will only recreate that once people figure out what's useful, what isn't and then we have FFXI all over again.
And I never said anything about healing jobs that heal the same way. I'd point to WoW and even Rift (though Rift for a while was iffy because of how people loved healing on Chloromancer) to show how different healing classes can work. In FF's case all we need is decide whcih job should do what. I'd personally suggest Geomancer (Half burst, half smart heals), Green Mage (75% HoTs, 25% burst heals) and Mystic/Oracle (single-target heals with additional effects granted by keeping debuffs on the mob). Then you have real choice as far as healers, and it may even introduce FF players to the concept of "tank heals" and "raid heals" assignments, which I hope are kept in mind by the developers for future content.