I can only comment on the first two games. I don't think I ever found power armor in New Vegas, and haven't played the other mainline games. I also haven't watched the TV series.
Power armor in the games does not have a specific weakness like what is shown in the question. Mechanically, all armor is basically the same; incoming damage is reduced based on the stats, and power armor has the best stats. The only unique feature of power armor is that it also boosts the character's strength. It is otherwise treated the same as a leather jacket, just with better numbers. The only downside is its weight.
Critical hits exist in the game - presumably they are modeled after the idea of "rare, lucky hits to a vital organ or the weakpoint of armor", but I don't think this is explicitly mentioned. However, one of the possible outcomes of a critical hit is to bypass armor - this will deal damage as if the target was wearing no armor, allowing a basic knife to deal damage even to someone in hardened power armor. This is entirely abstracted away to chance, and not to targeting a specific weak spot - even a bare punch dead center in the torso can bypass armor.
This is likely what the TV clip is demonstrating - an armor bypassing critical hit. I'm assuming that the person in power armor was not aware of the Ghoul and this would therefore be a Sneak Attack Critical, which is a guaranteed critical hit.
Lore wise, there is no mention of any weaknesses to the armor. The Glow has a spec sheet, which mentions the capabilities of the armor, and doesn't mention any weaknesses. Searching through the dialog files for "power armor" just shows that it is considered extremely powerful and valuable.
... other people started following us. Guess they figured if they weren't safe with a mutie and a Steel Knight [wearing power armor], safe just wasn't going to happen.
- Marcus's backstory in Broken Hills, Fallout 2
There's also a contemporary record in San Francisco stating that the Chinese PLA has no counter to the United States' T51-b power armor:
American military superiority now clear - T-51b power armor units deployed
to mainland. The PLA has no appropriate counter and is losing ground daily.