There are a few different types of scaling for your damage. I'll start out by showing you the (basic) damage formula:
(Flat damage × (i1 + i2 + i3 + ...) × m1 × m2 × m3 × ...) + (e1 + e2 + e3 + ...)
Where i represents increased damage, m represents more damage and e represents extra damage. The opposite of increased is reduced and the opposite of more is less - these are applied in the same way as their positive counterparts.
To start off with, you have your flat damage. Spells will have a flat, base damage value based on the level of the gem, while attacks will base their base damage on your weapon and then apply a multiplier to that. Other sources of flat damage include Abyss jewels (this league), Herald of Thunder / Ice, Wrath (for attacks only - it's a multiplier for lightning spells instead of flat damage), Anger, Added Lightning / Cold damage supports, etc. Notably, if a weapon says "Adds # - # [elemental] damage" then that is not applied to spells - it's a local mod that only applied to hits from that weapon (so attacks).
So your flat damage is the total of your spell gem damage / weapon DPS multiplied by attack multiplier (obviously attack-related stuff isn't relevant to you right now but it's good to know) plus all relevant sources of added damage. This is what is then scaled.
The first scaler is increased damage, and is what your question is about. Simply put, all sources of relevant increaseddamage are added together and then that number is multiplied by your flat damage. The relevant sources of increaseddamage will match with the tags on the skill you're using, so lightning spells will all scale with increased lightning damage, spell damage and elemental damage, along with any extra tags like area damage.
(A special case is Ball Lightning, which may or may not be relevant to you. It's tagged as Area of Effect but it doesn't scale with increased area damage. The area of effect is affected by increases to area of effect, though, and due to the nature of the skill its damage scales with the size of the area.)
What the increased damage specifically applies to doesn't matter, as long as it is relevant to your skill. 20% increased spell damage isn't worth any more or less than 20% increased lightning damage, for instance. However, 20% increased lightning damage will obviously affect lightning damage done by attacks, too, while spell damage won't.
Next up is more damage. Your damage is multiplied by each source of more damage separately, making it incredibly valuable. Usually the only source of more damage is support gems, which is why the more links you have on your skills the better. There are a few gems which apply more multiplier to skills that they're not linked to, which you'll often see in spell builds: Wrath, as mentioned, is a more multiplier for specifically lightning spells (with flat added lightning to attacks, as I mentioned earlier), and Arcane Surge, which is a durational buff that applies a more damage multiplier to all spells used, along with increasing you cast speed and mana regen. You'll often see Arcane Surge linked to Orb of Storms or Flame Dash as triggers; use one of those skills to trigger Arcane Surge and then your main skill will have a damage buff.
Then there's extra damage, which is applied after increased and more damage. This one is fairly self-explanatory: you take a percentage of one damage type and add that as another damage type.
That's quite possibly a lot more detail than you needed, but I wanted to make sure you had all the information. So to directly answer your question: yes, in your case, they are interchangeable. Stacking any relevant source of increaseddamage is fine because they're all added together.
