What was the initial purpose of true damage?

It exists so that some champions can specialize in fighting tanky enemies, or otherwise force tanky enemies to respect a true damage dealers threat rather than ignoring them because they bought a bunch of passive stats.

In other cases, it's for ease of use. Ignite is true damage partially so that everyone has an easier time judging when it will kill or not since it does the same damage to everyone (scaling on level, but still).

In cases like Ahri, the true damage is negligible but helps with her theme since she's throwing a soul stealing orb which seems like it should be more potent than just magic damage.

True damage, or it's equivalent in other games, is damage that makes an attack effective vs targets that build resistances to damage. Generally the damage would be less than an equivalent resist-able damage with the understanding that you would kill someone with resistances faster than someone with resist-able damage, but the target with resist-able damage would kill you faster than you could kill them with true damage.

Max %HP true damage is intended to make it so that an identical number of hits are needed to kill any target at a minimum, or to guarantee a kill at a certain health threshold.