The chord progression you're looking for is I–V–vi–IV.
In particular, the YouTube example you gave starts on vi, so the progression is traditionally annotated vi-IV-I-V. (I would argue that it's really a minor progression when you begin that way, so that it is i-VI-III-bVII, but that's just because I hear the i as a tonic. Your Mileage may vary.)
The Santiago song and the Sweeney piece are based on the same original melody, but that's not "Tha Mi Sgith" which is very different. The Sweeney piece is titled "Star of the County Down" which would indicates an Irish origin rather than a Scottish one. The Wikipedia page for "Star of the County Down" lists "Mädchen von Haithabu" as a recorded version.
These two progressions are different, although the second one is derivative of the first. As Ben mentioned, the first dm Bb F C, which is probably best interpreted as vi IV I V, which is a very currently popular progression.
The reason it sounds inspirational is that it encodes a narrative of striving against adversity.
dm: The initial chord is minor, ...
What defines a song greatly depends on the song. Some songs, like most fiddle tunes, are best defined by the melody. In other cases, a particular harmony defines the song better. What defines the song is really what sticks out to whoever is listening. This can be different between different people. Some songs are even so famous that anything can define them. ...
Perhaps the groove of the song. For example, some songs are made entirely out of percussion like for example Rebounds by Yannis Xenakis:
There is no conventional melody in this song, although what defines it is it's groove.
The end of this is a "perfect authentic cadence" which is considered the strongest ending cadence in classical music theory (V resolving to I, both in root position, melody ends on the tonic). That cadence can resolve either from above or below, but the resolution from below is particularly strong since it's considered a "leading tone", ...
What strikes me as most distinctive here is not the melody, but the structure, with the second half of the line repeated three times in total, and then a refrain. And the song that reminds me of is "This is the Way We Brush Our Teeth":
It's also the structure for "The Wheels on the Bus"
The track i was looking for is: "LCD Soundsystem - american dream".
I found a screenshot on my old phone that i have made when i heard the track the first time on the radio :)
Note: Answer created as wished by @chris-sunami-supports-monica
To my ear, it sounds like both songs on I'm Still Here were re-recorded using the same arrangements from Ten Thousand Angels. However, it's possible that, short of a complete re-recording, the backing tracks were remixed and remastered with additional instrumental tracks and new lead vocals. I lean toward complete re-recordings of both.
The differences tend ...