This is an argument of semantics, and one that is arbitrary. A “healer” is any “support” that restores HP. “Support” characters/classes have low overall damage in exchange for brining utility—non-damage related effects that make encounters easier: healing, mitigation, crowd control, mobility, cleansing, dispelling…

The only difference it would make if the role was labeled “support” is that we might’ve seen it also include jobs like bard, dancer, and red mage, and those jobs might’ve offered more support than just increased healing effects and a cleanse on bard, a little extra healing on dancer, and increased healing/mitigation and instant raises on red mage.