I just realized I explained a "Country Third" wrong to you in my post above.
A country third is a grace note that is one full step above or below (usually below) the main note you're going to strike after (or the same time) you release the country third key. A "grace note" is one semitone (half step) above or below the main note you're going to strike after you release that grace note. And when I say "release", I mean slide off or lift off that note. Typically your finger will literally "slide off" a grace note, and literally "lift off" a country third. Country Thirds and Grace Notes are very very quick notes that breathes life and feeling into a chord. I know you can hear them, because you're asking how it's done. Good job and good ears!
The song in this video above is using an out of tune piano which gives you the "honky tonk" sound along with some regular "grace notes". Because the keys are out of tune, it fattens up the sound similar to how a chorus pedal works. If Eddie Van Halen didn't detune one of his oscillators on his Oberheim in "Jump", that song would have been a flop. But since there was some detuning going on (which is normal in getting a fat synth sound), the song and riff are iconic even though it's "technically" out of tune. Tasteful detuning is a pleasant sound.
The song in your original post used "Country Thirds" and not "grace notes". Very similar techniques, but the country third will take a bit more practice to master.
fyi I am not a piano teacher, so excuse the crappy way I try to describe these techniques. I'll stick to selling carpet as my day job
