Yes, the tip has a step and is thinner. The video shows an ineffective tip that does not have the step, so all the tips I will be making this week have the step.

Meaning the tip will be thinner?

 

You could probably rewind a MOT (microwave oven transformer) with 1 or 2 turns of wire for your transformer. Or try the Weller soldering gun.

 

Meaning the tip will be thinner?

Yes, it will be thinner.

 

Yes, it will be thinner.

Good. That means it will have a higher resistance than the rest of the cylinder because it has a smaller cross section. That will concentrate the heat in the tip.
So if you have a long slot on both sides (can't tell from the pictures) the walls of the cylinder will just act like an extension of the wires to the tip where you want the heat.
Let us know how it comes out.
PS. Don't know if the cup in the tip thins the very end the same or not, but that would be nice, but it will probably heat anyway.

 

You could probably rewind a MOT (microwave oven transformer) with 1 or 2 turns of wire for your transformer. Or try the Weller soldering gun.

I've been doing some research on different types of transformers to see if there's a design that may help with my application (rather than just a standard laminated transformer). I have come across a few that sound like they may help, but I really am not sure.

Out of all the transformer variants, are there any that may help with my application? I was thinking about a high impedance transformer to reduce the inrush, and I'm also wondering how a toroidal transformer may impact the heating (the engineer working on the project before me tried to use toroidal transformers, but I'm not sure why he picked that style).

Another question, is there a way to vary the output voltage on the transformer other than having multiple taps on the primary? I'd like to vary the output voltage from 1.5V to 3.0V.

 

I've been doing some research on different types of transformers to see if there's a design that may help with my application (rather than just a standard laminated transformer). I have come across a few that sound like they may help, but I really am not sure.

Out of all the transformer variants, are there any that may help with my application? I was thinking about a high impedance transformer to reduce the inrush, and I'm also wondering how a toroidal transformer may impact the heating (the engineer working on the project before me tried to use toroidal transformers, but I'm not sure why he picked that style).

Another question, is there a way to vary the output voltage on the transformer other than having multiple taps on the primary? I'd like to vary the output voltage from 1.5V to 3.0V.

Either way is okay. Torroids are usually a little more expensive, but smaller. They do however: suffer from high inrush currents.
Your problem is how to specify what you want as it will be a custom one no matter which you choose.
What is it you don't like about switching the taps. It's probably the easiest. What would you like that you don't get from switching the primary taps.