You're half right. Nearly all commercial reactors are pressurized to some extent, but interestingly enough the Chernobyl and Fukushima reactors were not of the high pressure variety. Don't get me wrong, they're still under quite a bit of pressure, but no where near the 2000+ psi of a pressurized water reactor. So why did these reactors explode when no PWR has?
The Chernobyl disaster occurred because the Russian RBMK (a boiling water reactor, not a pressurized water reactor) reactors had a positive void coefficient. In a BWR the coolant (water) in its gaseous state is considered a void in the coolant. In most reactors designed with any lick of common sense, you try to make sure your design is stable by making your feedback coefficients negative. In most BWR's, the void coefficient is negative, meaning that as the void grows the power produced by the reactor goes down. Why is negative feedback desired? Well, as your reactor generates heat (power) it boils the water. If the reactor has a negative void coefficient, increased boiling from a power increase (lets say the boiling increases because you partially withdrew your control rods) will make the reactor naturally produce less power. As the reactor makes less power it will boil less water, and thus the power will increase again. The reactor will fluctuate until it stabilizes at some power level higher than before the power increase. If the reactor has a positive void coefficient, then increased boiling from a power increase will cause the power output of the reactor to increase even further. The same is true for a power decrease, as the boiling decreases the power tends to drop even further until (possibly) the reactor turns off. As you would expect, a positive void coefficient makes a reactor quite unstable. The void coefficient isn't the only feedback loop in a reactor, moderator temperature, fuel temperature, and other factors have their own feedback loops, but in the case of Chernobyl the void coefficient is the property of interest.
The RBMK design, interestingly enough had quite a large positive void coefficient. When the accident occurred, initiated by operator error, the reactor was making more power than it was designed for. As the power increased, the void fraction increased, causing the power to further increase. Under normal operation the automated control systems would counteract this by inserting control rods, but because of an experiment the operator had most of the control rods manually withdrawn. Because the automated control systems could not adequately counteract the positive feedback, the power increased very rapidly. This caused a lot of the water to flash to steam. Steam isn't an ideal gas, but anyone with a high school education should be familiar with PV=mRT, so you know that a real hot gas will be under high pressure. Things were made worse when the control rods were finally inserted in an attempt to shut down the reactor. The control rods were tipped with graphite (a stupid, stupid, stupid design flaw), and graphite is an excellent moderator (this basically means it slows down neutrons, and the more slow neutrons available, the more power a reactor produces), and actually caused the power to increase near the bottom of the control rods. All of this led to a MASSIVE power spike and flashed so much water to steam that the reactor vessel could not take the pressure, and that led to the explosion.
So you were half right in the sense that Chernobyl was a steam explosion, but the reason that the steam built up to such massive pressure was because of the positive void coefficient, not the initial operating pressure.
Most reactors are designed so that they have as much negative feedback as possible. This makes them inherently stable. If your all of your reactivity coefficients are negative (a good design), you don't need to worry about the power your reactor produces spiraling out of control. What I am saying is that high pressure reactors can be, and in most designs are, self stabilizing. Interestingly enough however, LFTRs actually have a slightly positive moderator temperature coefficient.
Now what about the fukushima accident? Well, the fukushima accident wasn't a pressure explosion at all. It was a hydrogen explosion. When the earthquake hit japan, the reactor successfully shutdown, but as with all reactors it continued to produce heat because of radioactive decay from short lived isotopes. This wasn't initially a problem because nuclear power plants have backup systems to continue operating the coolant systems to keep the reactor cool. Unfortunately when the tsunami hit the power plant, it disabled the backup systems. When the coolant systems no longer had power, the fuel rods (which were still hot from radioactive decay) began to heat up and the cladding began to react (chemically, not nuclear) with the water and produced hydrogen gas (this is very explosive stuff). This hydrogen gas was eventually ignited and this is what caused the explosions, not a pressure build up.
Now, the fact that LFTR's operate at nearly atmospheric pressure is a neat safety feature, but to say that high pressure reactors are inherently unstable is just wrong. I'm all for LFTRs, but keep in mind that what you are listening to is essentially a sales pitch. He's trying make a strong distinction between molten salt thorium reactors and the negative image that light water reactors have (quite unfairly) received over the years. Molten salt reactors aren't without their own unique challenges.
Again, let me reiterate that I'm all for the development of LFTRs, but the way you have portrayed light water reactors as being unsafe isn't true at all. The overwhelming majority of reactors are light water reactors, yet nuclear power has claimed fewer lives per KWh than any other form of electricity. That includes wind, solar, and hydroelectric.