In my mind the terms "primary" and "backup" mean something different than what they used to. It used to be that "primary" meant a canister light and "backup" meant a handheld light.
To ME, NOW, primary means a light that I'm going to turn on and keep on for the whole dive. Backup means a light that I will only use "as needed".
Most dives do not need the burn time of a canister light. Really, I think it's only long cave dives are likely to need a can light. Even deep, technical wreck dives don't generally need the burn time of a can light. Bottom time is short and the light is not needed (usually) once you are shallow enough to be doing the longer deco stops.
So, for a handheld primary light, I want:
A light that takes 1 x 26650 battery. Gives longer burn time than an 18650 and only having one cell means the light should not be so long that having it on the back of my hand is annoying.
Has multiple brightness levels. Especially important on night dives.
Tight spot beam. 6 - 8 degree angle for the hot spot of the beam.
My personal favorite for this is the Xtar D26 Whale. 1100 lumens (rated). Long burn time. Mechanical lock (quarter-turn) on the On/Off button. However, the On/Off lock is not AS important on a Primary, since the premise is that you'll turn it on and leave it on during the dive. You wouldn't have the concern of having it clipped off or in a pocket and getting turned on and burning battery without you realizing it.
For a backup light, I want:
A light that takes 1 x 18650 battery. The light is less bulky, so nicer to have stowed somewhere. I keep mine clipped to a D-ring on a shoulder strap, with some bungee holding the other end to the shoulder strap down lower.
Multiple brightness levels not required. It's for "as needed" use, so a single brightness is fine, and brighter is better. 1000-1100 lumens (rated) is fine.
Will NOT turn on by accident. A twist on is ideal. A tail on/off button is good. A side button would have to have a mechanical (versus electronic) lock and even so is not optimal (to me - for a Backup).
I have 2 x DGX 600 Twist lights that are about 6 (or 7?) years old and still work perfectly. I keep at least one clipped to my harness at all times. 2 on night dives.
The DGX 600 is only claimed to be 600 lumens, but it's the same LED emitter as all the 1000 - 1100 lumens (claimed) lights use - a Cree XM-L2 (U2). In my experience it is basically just as bright as any of those. DGX is just a little more "honest" about how many lumens it produces. Their notes do say that it is 1100 "raw" lumens.
For someone just going to do their Advanced training, I think 2 x DGX 600 Twist lights are a good place to start. They are very inexpensive - as dive lights go. They will get you by for anything you need until such time as you determine you have a specific need for something different (like, maybe a light with multiple brightness levels). And DGX has The Best return policy in the industry. Basically, it is unlimited. You could return a product (with some exceptions) to them after using it for 5 years, if you wanted to, for full credit of purchase price. Unused and within 30 days will get you a full refund, if you want. Used, or after 30 days, you get store credit, but it is still for the full purchase price. It is a policy that is wide open for abuse. I can only speculate that if someone did choose to abuse that policy, DGX would simply stop selling to them. Hopefully, no one will abuse them to the point of ruining such a generous return policy for the rest of us.
I am not specifically recommending the DGX 600 Button light, because I SUSPECT that it would be too easy to turn on by accident when clipped off or in a pocket somewhere. However, a Button for Primary and a Twist for Backup would not be a bad plan, if you wanted to go that way. The Button would then give you multiple brightness levels to choose from.