Are AA battery and 18650 battery the same?
I have an Antex SZ004W0 wireless soldering Iron which uses three 1.5V, AA batteries, but I don't know if it supports rechargeable batteries. I know that it supports non-rechargeable battery named "Kodak Max LR6" because I tried.
When I tried to find the best rechargeable AA battery on Amazon or Ebay I got a 18650 battery) which has dimensions: 6.5 x 1.8 cm.
Based on the dimensions being different, I think that AA battery and 18650 battery are not the same.
Which one can I use in my soldering iron?
It's obvious that the 18650 isn't the right choice due to the output voltage which is more than twice the output of the most common alkaline batteries your soldering iron needs 4,5 volts and will get 10,4 volts using the 18650 batteries.
LR6 means AA alkaline, regular Duracell or Energizer (or store brand etc) would be equivalent to the Kodak. The Energizer L91 in your second link is intended as a drop-in replacement for alkaline which uses lithium-based chemistry instead. The voltage is slightly higher (maybe 0.2V higher or so when brand new) but they will work fine in most, perhaps nearly all, products which can run from alkaline. They have 2-3 times the energy storage at lower drain rates and may provide much more than that compared to alkaline in high-drain applications. They cost more and aren't rechargeable though.
An AA cell is not the same as an 18650 cell. 18650 designates lithium-ion batteries of nominal voltage 3.6 Volts each. These typially consist of the actual lithium-ion battery and some circuitry, as shown in this image:
Primary (non-rechargeable) AA cells are typically around 1.5 Volts each, such as the classic Zinc-Carbon batteries, as well as some Alkaline batteries. Rechargeable AA cells have nominal voltages that vary with the chemistry: Nickel-Cadmium and Nickel-Metal-Hydride ones are around 1.2 Volts each, Nickel-Zinc are around 1.6 Volts.
In brief: No, the 18650 cells can not be used in place of standard AA batteries, rechargeable or otherwise: The nominal voltage of the 18650 is significantly different from the 1.5x3=4.5V of the three AA cells, assuming those are connected in series. In case the AA cells are connected in parallel to supply sufficient current to heat the element, that's worse, as the voltage of the AAs would be 1.5 volts nominal.
So what do you suggest? Should I buy AA battery for my Soldering Iron?
The Antex SZ004W0 is designed to work with AA cells, and will kind-of work with NiCd rechargeable AA cells as well. The time to heat up will be greater, and the batteries will last a very short while before needing recharging. You might do better by using Alkaline AA (non-rechargeable) cells instead of regular Zinc batteries.
Note that the original poster said it takes *three* AAAs, that would be *more* voltage than the 18650 if the AAAs were completely new.
If you have a physical adapter case, 1 X 3.6 V lithium ion battery may suffice your application although it will be weak compared to 1.5V*3=4.5V. On the other hand, it will be as strong as 3 X 1.2V AA rechargables. Also, keep in mind that a 3.6 V rechargeable lithium ion battery actually gives an output starting at 4.2 V and ends at 3.0 V, so it likely won't perform as bad as you expect. It's also rechargeable which will save you lots of money in the long run.
Also, do NOT buy any kind of UltraFire battery. They're known to have lots of copycat manufacturers which relabel batteries incorrectly. There's no such thing as a 5000mAh 18650 lithium ion battery. Anything above around 2400 or 2600 mAh should be thought of as suspect. People who've tried these 4000-5000 mAh batteries have found them to die early and have less juice than a 1000 mAh battery.