My instinct would be that food will cook quicker just before the water starts boiling because steam conducts heat slower than water and so when the food is in contact with the steam bubbles it receives less thermal energy.
This is absolutely the opposite of reality!
A steam bubble in contact with a body with a temperature less than
100°C will condense on this body, transferring about 540 cal/g!
In fact heating a wall by condensing steam is one of the most
efficient heat transfer methods.
But now your primary question: those noodles "cook" faster
where they experience the higher temperature. In a pot on a fire,
the surface of the water will be as low as maybe 80 or 90 degrees,
when bubbles raise at the bottom, but do not reach the surface yet.
(Again here: heat transfer by bubble evaporation is extremely
fast, same reasons as for condensation)
10 or 20 °C less are a lot when cooking noodles (wheat starch)
this may lower the reaction rate to half the value at 100°C,
or even lower (rule of thumb for such reaction).
Thermal conductivity: most amorphous solids/liquids have similar, low
heat conduction. Crystalline solids are medium, metals are much higher,
the thermal conductivity being related to the electrical conductivity.
Google for: Wiedemann Franz Lorenz.
Thermal conductivity of gases can be calculated using kinetic gas theory,
this was "triumph" for Clausius, Maxwell and Boltzmann.
To start with, "water freezes faster when it starts out hot" is not terribly precise. There are lots of different experiments you could try, over a huge range of initial conditions, that could all give different results. Wikipedia quotes an article Hot Water Can Freeze Faster Than Cold by Jeng which reviews approaches to the problem up to 2006 and proposes a more precise definition of the problem:
There exists a set of initial parameters, and a pair of temperatures, such that given two bodies of water identical in these parameters, and differing only in initial uniform temperatures, the hot one will freeze sooner.
However, even that definition still has problems, which Jeng recognizes: first, there's the question of what "freeze" means (some ice forms, or the water freezes solid all the way through); second, the hypothesis is completely unfalsifiable. Even if you restrict the hypothesis to the range of conditions reasonably attainable in everyday life, to explain why the effect is so frequently noted anecdotally, there's literally an infinite number of possible experimental conditions to test, and you can always claim that the correct conditions just haven't been tested yet.
So, the fact that the internet is awash in a variety of different explanations makes perfect sense: there really are a bunch of different reasons why initially hotter water may freeze faster than initially colder water, depending on the precise situation and the definition of "freeze" that you use.
The paper you link to, O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox by Zhang et al., with results echoed by the HowStuffWorks video, attempts to solve the problem for a very specific sub-hypothesis. They eliminate the problem of defining freezing by treating freezing as a proxy for cooling in general, and directly measuring cooling rates instead. That experimental design implicitly eliminates one internet-provided explanation right off the bat: it can't possibly be supercooling, because whether the water supercools or solidifies when it gets to freezing temperature is an entirely different question from how quickly it cools to a temperature where it could freeze.
They also further constrain the problem by looking for explanations that cannot apply to any other liquid. After all, the Mpemba effect is about why hot water freezes faster; nobody is reporting anomalous freezing of, say, hot alcohol. That might just be because people freeze water a lot, and we don't tend to work with a lot of other exotic chemicals in day-to-day life, but choosing to focus on that restriction makes the problem more well-defined, and again implicitly rules out a lot of potential explanations ahead of time- i.e., it can't have anything to do with evaporation (because lots of liquids undergo evaporative cooling, and that's cheating anyway 'cause it changes the mass of the liquid under consideration) or conduction coupling to the freezer shelf (because that has nothing to do with the physical properties of the liquid, and everything to do with an uncontrolled experimental environment, as explained by John Rennie.
So, there really isn't just one answer to "why does hot water freeze faster than cold water", because the question is ill-posed. If you give someone a specific experimental set-up, then you can get a specific answer, and there are a lot of different answers for different set-ups. But, if you want to know "why does initially-hotter water cool faster through a range of lower temperatures than water that started out at those lower temperatures, while no other known liquid appears to behave this way" (thus contributing to it freezing first if it doesn't supercool), Zhang has your answer, and it's because of the weird interplay between water's intra- and inter-molecular bond energies. As far as I can tell, that paper has not yet been replicated, so you may consider it unconfirmed, but it's a pretty well-reasoned explanation for a very specific question, which is probably an influencing factor in a lot of other cooling-down-hot-water situations. There is a follow-up article, Mpemba Paradox Revisited -- Numerical Reinforcement, which provides additional simulation evidence for the bond-energy explanation, but it can't really be considered independent confirmation because it's by the same four authors.
Best Answer
There is a reasonable chance that your tea is cooling faster.
This is somewhat counter-intuitive, but it's because the only variable here isn't the temperature.
Evaporation is also very important in the cooling of the tea. The kettle probably has much less available area to release the steam, so the air above is saturated. This will lead to a much slower rate of evaporation, compared to when the boiling water is exposed to open dry air. This would mean if you poured the same amount of water into each tea, yours would lose more mass through evaporation which could be the deciding factor.
The rate of evaporation depends on the surface area and vapour saturation in the air. A kettle has a small opening to exchange it's saturated air with the surroundings, while the open cup can easily move away the saturated air through natural convective currents and new fresh air can come in to take it's place.
If you put some sort of loose covering on your cup that emulated a kettle while it cools for the first few seconds (be careful to make sure steam can still get out, just not as easily) you should see it remain quite a bit warmer.
Edit: Adjusted the answer to reflect brought up in Vectorjohn's answer. I was getting too invested into evaporation and lost track of why I brought it up in the first place. It changes the mass of the tea you have to cool if you pour both to the same level.