Seedlings have enough for the first week or more of growth, but it definitely needs a full diet after that... and there are benefits to providing that from the get-go, too, even if the seedling doesn't need it -- obviously "too much" is always bad, too.
Are these tiny plant sin a huge pot? could be the result of the difficulty to properly water a tiny plant in a huge mass of substrate. Make suer you water a column deep down, preferably all the way down to bottom of the pot.. outdoors you'd have to consult with an outdoor grower on depth.
partially watering a pot, more so depth than width, leads to more than 1 potential problem. with a tiny plant you are forced to water a smaller diameter aroundthe plant or it'll stay wet for weeks, which will just grow nasty pathodgens. If yo don't water all the way down, you'll promote superficial roots, which is not optimal. The dryback pattern may cause large portions of roots to dry out too even though the top of the soil is still moist.. also not optimal.
the super pale, undeveloped seedling looks like it doesn't have a growth tip. if that is fresh from a shell, it may take a second to get a green pigment going, but if it's been above ground for a while, i'd be a bit concerned about it. It's certainly not peak health.
How you fertilizer and when depends a lot on the soil you are using, soss that's soemthing you have to work out with trial and error. the new growth looks okay and if the damage stays isolated to the cotyledons and first single-bladed leaf, i wouldn't worry about it too much. Maybe an initial root wrapped itself around some pre-amended nutes and singed itself, if the soil is "hot" this is possible. somethign drip on it?
either case, more than one thing causes that sort of tip damage, so you may need to see it progress more to properly diagnose with confidence. And, if it doesn't progress, you saved yourself the mistake of adding something unnecessarily. It's always easier to add things than to take away.
I doubt eggshells and terpinator, whatever that is, is a proper diet on its own. 0-2-4 so, no that is not a full diet. When the soil runs out of its precharge, you'll definitely need more.. something with nitrogen for sure...