Magic: An ability that is defined to a certain property such as fire, electricity, light, etc.
Magic tree:
Attack Magic: Anything that is confined to a specific property AND is meant to deal damage
Heal Magic: Anything that will result in a positive effect on your party members (excluding cure magic)
Skill: A character, or weapon specific ability that isn't necessarily confined to a single property however is most frequently a physical attack.
Skill Tree:
Attack skills: These will deal out mainly physical damage and may or may not have an attribute associated with them.
(De)Buff Skills: Anything that increases (or decreases) the ATK, SPD, or DEF of a target
Status Skills: A skill that inflicts a status ailment on said target. (i.e "Old-One" or "Balzac")

Magic: An ability that uses SP, and is defined by association with various elements (fire, earth, wind, water).
Magic Tree: A group of closely related magic abilities.
Spell: The manifestation of magic.
Offensive Magic: Spells designated to cause damage to an enemy.
Supportive Magic: Spells designated to assist teammates or heal wounds.
Skill: An ability that uses SP, and is defined by having no definite association with an element. Is almost always a physical attack.
Skill Tree: A group of closely related skills, confined to a single weapon type.
Offensive Skill: A skill intended to cause damage to the enemy.
(De)Buff Skill: Skills defined by their ability to support the party with buffs, or debilitate the enemy as debuffs (anything that lowers a stat).
Status Skill/Magic: Skills or magic defined by their ability to inflict status on the target.