Initiative really doesn't matter. Setting aside the Ready conversation, if I am a mage and cast Hold Person on an orc, he will get 2 saves before the start of my next turn. If I won initiative it goes as described above. If the orc won initiative then he does something then when my turn comes around it still happens as described above.
It matters because there are other PCs in the party. It's not your initiative relative to the orc that matters, but your initiative relative to your rogue and fighter comrades.
Let's assume the orc fails its first save and makes its second save. Here's one possible initiative sequence:
Fighter: Attacks the orc normally.
Rogue: Attacks the orc normally.
Wizard: Casts
hold person. Orc fails its first save and is paralyzed.
Orc: Loses its action, makes its second save. Orc is no longer paralyzed.
Fighter: [combat continues]
Compare to:
Wizard: Casts
hold person. Orc fails its first save and is paralyzed.
Fighter: Attacks the orc with advantage for an automatic crit.
Rogue: Attacks the orc with advantage for an automatic crit, including double Sneak Attack damage.
Orc: Loses its action, makes its second save. Orc is no longer paralyzed, but is probably dead anyhow.
Wizard: [combat continues]
The fighter and rogue get a big damage boost in scenario #2 compared to scenario #1. It's to the wizard's advantage to arrange things so the fighter and rogue get a chance to strike before the orc has a second chance to save. This can be done either by the wizard readying till after the orc has acted, or by the fighter and rogue readying till after the wizard casts.
Both of these have tradeoffs, of course. If the wizard readies, the spell could be disrupted, the orc gets a round of attacks in before the paralysis hits, and the wizard gives up the ability to cast
shield for a round. If the fighter and rogue ready, they sacrifice their ability to make OAs for a round. (Overall, I would say it's almost always better to have the fighter and rogue be the ones to ready attacks. However, it can be hard to coordinate that kind of thing.)
Has there been any clarity provided to intent of "Ready"? It certainly (and fairly) implies a reasonably specific trigger. But in this specific case "I will Ready to cast Hold Person as soon as the orc does anything" seems reasonable. It is outside of the larger general concept of Ready, but it doesn't seem to be against the spirit. From a pure concept the idea that I'm paying attention to this specific character with intent to lock him down, is reasonable. So it isn't an exploit to apply a broad interpretation here as an exception. (IMO AKA "good DMing").
As with so many things in 5E, the wording leaves it pretty open to interpretation. However, readying for "when the fighter attacks" is indisputably a valid trigger, and would also achieve the goal.