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That's why paladins need STR (or AGI) for melee hits AND CHA for spells targeted at enemies, and that's also why people love to multiclass Haxblade (or, in BG2, pact of the blade warlock) to be able to use CHA for their weapon attack rolls.
The spell will call for what type of save (if any) that is required, and it is calculated off your primary casting stat. If a spell calls for a Wisdom save, and you are a Wizard, then only your Intellect is used to calculate how hard it is to save against the spell.
The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.
Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
Most spells that require attack rolls involve ranged attacks. Remember that you have disadvantage on a ranged attack roll if you are within 5 feet of a hostile creature that can see you and that isn't incapacitated.
Smite is an ability that consumes spell slots. Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend, to a maximum of 6d8.
So if, say, a smite spell has a constitution save, and an enemy has 8 constitution, they'd roll 1d20-1 for their save, against your CHA based save DC.
The smite spells you are using are triggered on a successful attack, have guaranteed bonus damage and a chance to proc an effect.
The save, is the ability score your enemy uses to resist the effect. For example a stength save, causes the enemy to roll a D20 + their strength modifier to resist the effect.
The DC (difficulty check) the enemy has to overcome is 8 + proficiency (2 @ early levels) + your spellcasting modifier (charisma +2) = 12.
I'm not sure what the -1 you are referring to is.
they're talking about divine smite, you are correct in that thunderous smite is a spell and here's how it works:
you use a bonus action to cast thunderous smite, you then use an attack to deliver the spell. if the attack hits, you do the attack's regular damage plus the spell's damage. at this point, the target has to make a strength saving throw against your spell save DC or be knocked prone. so your DC is determined (8 + proficiency + charisma bonus in a paladin's case) so it's 8 +2 +2 = 12. the target would roll plus their strength modifier and, if they're proficient in strength saves, that bonus too, to meet or beat a 12.
so when you report that the game is showing your spell save as being -1, it's confusing because that number cannot possibly relate to that stat. it's either a display bug or there's a misinterpretation happening
edit: it could be showing that the TARGET is saving at -1 due to a low strength stat
EDIT: And yes, each class has a different casting ability. Wizards, Eldritch Knights (a fighter subclass) and Arcane Tricksters (a rogue subclass) use intelligence; cleric, druids, and rangers use wisdom; bards, paladins, sorcerers, and warlocks use charisma.
I would double check the combat log after using a Smite to see if the correct number is being used. Might be a visual bug, but it could also be a bigger issue.