What's the other end look like? What standard 14500 did you try? Am I correct in assuming that you are assuming both the positive and negative contacts are on the top of the cell?!
If I put any standard 14500 in it, the led just slights lights up.
Isn't that the idea? If it worked at all, it's a 14500 light. and don't be fooled by that weird looking flattop. It's probably just more area for contact, which means if a smaller flattop surface isn't connecting, placing a thin coin-shaped conducting metal on top (like a thin, neodymium magnet wider than the contact) will help make full contact. That cell is probably protected so it is a little longer, meaning maybe the spring is crushed a little, so a little extra to help make full contact should do the trick. But I really think if the LED turned on at all, it was the right cell with a bad history or a crummy cell.
This is the light?
I don't think there's anything special about how that light makes contact, which is to say, using a spring that contacts the positive end of a 14500. I have Thrunite AAA lights, and they all take bog standard AAA and 10440. I'd say it's far more likely there's something wrong with the other "standard" 14500 you tried. Poor quality cells and abused cells do not put out any amps nor have decent capacity left in them.
I'd recommend you try
a decent 14500 cell. I've been running these constantly since Nov. 2020 for 1.2A and 1.8A lights, and they are awesome. Lithium nickel rechargeable or INR cells use a LiNiMnCoO2 nickel-manganese-cobolt oxide (NMC) hybrid chemistry for increased energy capacity.
Here is HKJ's test/review.
Splurge and get a quad, with ship will be just under $20 if you choose the USPS option. If you treat these cells right, they will last. I wouldn't use the builtin charger to charge the cell. A decent Li-ion charger can be had for under $20 with a digital readout showing voltage and adjustable charging amps.
Forgive me if this is patronizing, but always rest your Li-ion cells after use and before charging, and after charging and before use. An hour is fine, but they just have to cool down, so in cool ambient temperatures 10 minutes can be fine. Actually, heat is the enemy, so when running long periods, try to give them a chance to cool down before continuing, though I haven't noticed any issues running them from full and rested to 3.6V empty at high amps and getting too hot to hold. And avoid dropping them, that also can reduce capacity. That is really the hardest thing, because they love to slip out of my fingers, forcing me to move faster than I like to catch them before they hit anything. Gripping them tightly helps.
That is a neat light. I think I need one of those.
Reading a little more about the Archer Pro, apparently the cell is not meant to be replaceable. But you got it open, so why not? The switch concerns me. Does it have a sleeve inside the body that moves forward when you press the switch (ala Lumintop FW lights)? Or does it feel like a standard electronic switch, with no mechanical action? idky it would matter. But it is possible ThruNite for some reason designed a light with built-in obsolescence with a weird cell. I am curious if you have a way to test the voltage on your original cell. Does the bottom have a negative contact? Then if Vapecell H10 doesn't work right away with no modification, I think the coin magnet trick should work if the spring needs more contact surface. But if the negative and positive contact are both on the top of the cell somehow and for some reason, that light is a junk design. I'd find it hard to believe.