Incantation and English name
Each spell has two names. The incantation is the short foreign word a character calls out. The English name is what the rest of the story uses. Glacies is Latin for ice, and a Lance shoots a sharp spear of magical energy, so Glacies Lance is the Ice Lance.
The roots are single words, not full Latin sentences. Lux is light, umbra is shadow, ignis is fire, vincere is to conquer and protegere is to protect. The tool joins them for style, so treat the result as flavor, not as correct grammar.
English endings name the effect. Veil hides the target, Nova is a sudden bright burst, and Requiem is a peaceful spell meant to put spirits to rest.
Schools of magic and a trap in the names
Use the school and element pickers to narrow the list. Many school names end in -mantia, which shares its Greek source with the English ending -mancy and means divination. So Pyromantia is divination by reading flames, and Hydromantia is seeing visions in still water. Neither one throws fire or controls water.
Other schools are about action. Goetia is dark magic that summons and commands spirits. Theurgia is white magic that asks angels and good spirits for help. Alchimia changes materials. For a fire attack, pick the fire element and a root such as flamma, a softer and more focused flame.
The mana circle ladder
Many Korean-style fantasies count a mage's power in circles of mana around the heart. The ladder panel runs from novice mage (1st to 2nd Circle) through combat mages (3rd to 5th) and high mages who can fly (6th to 7th), then the sage at the 8th Circle who can change weather, up to the 9th Circle, a transcendent master. Check the spell against the ladder so a 3rd Circle mage does not cast sky-sized magic.