Late to the party but hopefully able to help out, I've replaced many of these during my years doing contractor work. I'm extremely happy that you've decided to tackle this on your own, it's great for learning about the systems in your house and saves you a ton of money.

The reset button doesn't always pop (it's for overheating), but your description of having room temperature water probably means one of the elements aren't working (one that's got both completely out will be the same as your cold water running). This is assuming the water gets used pretty frequently as is suggested with a two-person household with showers/kitchen/etc.

As you see from this diagram there is a "dip tube" that the cold water comes in from, the elements will heat the water and make it rise to be the first water to exit from the tank. The rising water without both elements heating it will slowly dissipate causing it to continuously cycle the water up and down as it heats and cools.

You're on the right track with the videos you posted but kinda veered off from instructions with your testing. Continuity testing isn't a sure-fire accurate assessment unfortunately, also you're not placing your probes in the right spots for testing as is recommended in your video (timestamp around 6:30). No-contact voltage testers are.... iffy at times. It's basically just used as a safety measure to double check yourself before you start messing with stuff. I've gotten false positives plenty of times over the years with different devices. But, never just go with it, check over everything again including physically checking electrical line runs to make sure someone didn't splice something in that's about to shock you.

If you're a tinkerer like me, get familiar with your electrical diagram for your tank. It will be one of the two following configurations,

That silver conduit piping on the top of your tank is the electrical line-in. If you want to completely eliminate the breaker/electrical line from being a problem you can unscrew the electrical panel on top (shut the breaker off first) to see the wires spliced inside (more than likely just wire nuts). Always use safety first, shut off the breaker and test the top with your no-contact before you start un-nutting wires. Take pictures of the wires if you feel it's necessary to know exactly what is connected where (good practice to keep yourself from being screwed on bigger projects and something to look back as a "I did that"). Make sure the wires are secured in a manner that will not electrify the tank or anything around it before turning the breaker back on to test it. If those wires are good you know the problem is with the tank. Pre-twist the wires together before putting the wire nuts back on. It can be daunting, but so is automotive or even working around an oven, precaution and awareness, you've got this and don't rush ANYTHING.

If it's all good, I would suggest buying the kit from home depot if it's not something that's gonna hit your wallet too hard. Worse case scenario, you don't need it and eventually you WILL need it since all water heaters eventually break (the tank is just an insulated tank, the thermostats and elements are really the only "thing" that can break). 10% of the time it's a thermostat, it's very rare and usually from the environment like being installed in a crawl space (corrosion and etc). 80% of the time it's an element, other 10% will be a leak (again usually from the condition of the space). Your setup looks very well conditioned for your tank, so I would bet on an element.

Things You'll Need:

• Phillips head screwdriver
• Socket for the element and breaker bar (or big-ass wrench with a metal pipe setup if it's been ages since it was opened or just an element wrench if it's nice to you, sockets are better if you're looking at future projects since the element wrenches can be dainty )
• Garden hose long enough to drain the tank outside at a lower elevation (gravity drain)
• Flat head screwdriver large enough to comfortably turn the drain valve below

The usual school of thought is if you're replacing one thing, just replace all of it. Since you're the worker and home-owner it's up to you. I have had kits with bad parts, but it's rare so just keep that in mind.

I found a pretty basic manual from state selects that will cover your model, seems there are some shenanigans with how they do their specific model numbers (it happens). Bottom of Page 15 begins the process for thermostat and later on element removal/inspection/replacement. You can inspect the elements first and replace the one probably bad one and see if that fixes your problem with spare parts left over for a future fix, or just go to town with it all.

Some quick side notes:

1. turn the hotwater tank breaker off and run the warm water out with a sink tap before you shut off the cold water valve and start draining (so incase there's a pocket of scorching water it's not an issue).
2. After closing the cold water valve to the hot water tank, leave a hot water sink valve open to allow air in and completely drain it
3. There are other things to do "maintenance" wise if you're up to the task and already balls deep in the tank at this point:

• Check the water draining to see if you have debris coming out of the hose (should be flushing your tank periodically but most homeowners don't, reduces capacity and efficiency so you could be making your tank even better than it once was). If you see debris just keep the cold water on till it stops.
• Check the anode rod if you're unsure of it's condition, it's a sacrificial rod inside of your tank (the manual discusses it as well, top of page 15)
• test the Temperature-Pressure Release valve, again page 15 (TPR is not the drain valve, see diagram at the top with the dip tube)

Continued side notes/concerns:

• Those wires running across the floor scream at me, right near the discharge hose spot as well with oldschool wire staples without the protective rubber. I'm assuming the black wire runs behind the tank and up into the conduit (they atleast put conduit on but seems kinda silly with the rest ran like that). Just be careful, one earthquake and some water leakage could mean electrified area if the wires get damaged. Re-run those lines if/when you feel like it.
• Is there a pan under your tank? It's kinda dark so couldn't tell, makes me concerned for where the TPR valve is being ran too and I hate when they install the tank so you can't see it (probably on the right side under the insulation). If you don't have an expansion tank on your line then you could be getting drips where you can't see (more often it drips somewhere that a shitty valve is installed like a toilet). If there's no drip pan, then if the tank is damaged the water will go wherever it wants but the TPR should have a pipe running into a crawlspace or outside depending on local code. If the valve is just sitting there, with no pipe coming out, you have another project that needs addressed because it's a ticking time bomb that will flood one day. Best bet is to have a drip pan that's ran out and the TPR valve with a pipe to the drip pan so it's all caught.

Good luck, I'm here if you need anything additional!

Does the heater have its own breaker, and is that breaker single or double pole? (does it take up one slot ot two in the pannel?) I was asking to see what voltage to expect. Looked at your previous post; it's a 240vac heater.

Connect one probe to a known good ground. (the water pipes should be grounded if metal, the case of the heater should be grounded as well) Probe each of the terminals with the other probe to find which wire is live. (checking for voltage)

Here's a typical diagram.

Those top two terminals you are probing should be live1 and live2 (split phase AC). Each one should measure ~120vac to ground, and there should be ~240vac between them. If either wire is broken, you'll get 0 volts between them and maybe one still showing 120->ground.

Based on your non-contact tester going off, but 0vac between those top two terminals: I would say there is a broken/disconnected live wire in between the heater and the breaker supplying it.

Is there perhaps a light switch nearby that doesn't seem to control any lights? Sometimes the water heater is wired to a switch, so it can be turned off easily while away from home for an extended period.

Any chance your control wiring is DC?