Is it safe to touch 200 volts DC?

No, most electrical regs class low voltage “safe” as below 50V DC

It is not the max current but where it travels through the body and, also, when in the heart cycle it hits...

So, don’t play.

The content maker simply declared that 200/300 volts DC is safe to touch.

He didn't say this anywhere in the video, and he didn't imply it either. In the first few seconds he says:

I want to kick it up a notch and show you at what voltage it hurts over skin

He is just showing "when does it start hurting". That's all.

He has another video of a live demonstration, hooking up multiple car batteries to produce 120V DC and touches them: https://youtu.be/ZxBF7WC0TQk?t=654

He feels it and it hurts, but not nearly as much as 120V AC.

He's not saying DC is always safe. He explains the dangers in a different video where he talks about the relationship between voltage and current in hurting you: https://youtu.be/XDf2nhfxVzg

The linked video at 3:48 reveals something very essential of "teacher's" capabilities. He can output nonsense with quite assuring voice. I do not think he believes himself that human body has capacitance which makes easier for AC voltages than for DC voltages to generate harmful current.

Do not believe him. Even much lower DC voltage can be dangerous. I have seen how a car battery made in about 1 second a screwdriver so hot that it burned its way to user's flesh. 100V or more DC can stop ones heart when a person gets it from hand to hand or from hand to feet. There's no exact voltage limit for this. Someone can stand more than another. The moisture and how tight the contact happens to be effects, too.

Regulation codes for electric works vary. Check which is the limit of extra low voltage (=ELV) where you live. Staying below the limit does not make the voltage safe, but it's used in a court of law when the judge must decide if someone has caused a danger intentionally. Any electric works with higher than ELV voltages can be considered intentional searching for troubles, no matter did a person know that such limit exists or not.

AC causes pain continuously because the current turns forth and back and every pulse disturbs the nerve system. DC hurts that way only when the current starts or stops. But high enough starting pulse can stop one's heart.

"Teacher's" circuit was galvanically connected to mains AC. His touching the DC output could have given a different result if he had accidentally a connection to the ground for example through leather shoes and a concrete floor. Having a rectifier would in that case make no difference.