One thing you might be hearing, and liking, is a different type of tube distortion. Tube amps have two stages, pre-amp, and power-amp. Both stages can distort, but they have very different characteristics.
Pre-amp distortion is what we normally associate with a "fuzz" or rock sound. It's usually very distinctive and obvious. It doesn't have to be extreme, but it still isn't the same sound that you'll get from the other stage.
Power-amp distortion seldom sounds like a "fuzz" or distortion box, and is more often associated with blues guitar. Its effect on the guitar's sound adds compression and fattens the sound. It's a hard effect to get on a lot of amps, because they have to be very loud to get it; The power-amp section has to be pushed hard to start saturating the tubes enough to get the tone, though some of the new 1 watt and 5 watt tube amps can do it at reasonable volumes.
Egnater amps have a wattage setting that controls the voltage going to the tubes. By reducing it the amp's output drops. It doesn't change the volume or tone though, instead it changes the point that the output tubes begin to distort, letting you get power-amp distortion at a lot lower volume. I have their Rebel 30, and that ability was one of the main reasons I bought it - at just a little over conversation level you can hear the power tubes starting to distort. It's a very clean sound with natural compression - a singing tone. Other amps can do it too, especially those 1 and 5 watt tube amps. Class A amps can get a fatter sound along the same lines because of how they use the tubes.
In addition, there might be a bit of reverb on the guitars in the first two mentioned videos. It's not necessary to use a lot with output-tube distortion but it softens up the sound and adds a touch of that singing sound. Listen to older Santana solos, especially when he's got a singing sustained tone and you'll often hear his reverb smoothing it out.
The third video doesn't sound the same to me, but then again, it's a jazz-chorus amp, which is solid-state. It's voiced to be a mellow, super clean amp with a solid-state distortion, but can't do the output stage distortion because it doesn't have tubes - and you don't want to hear how power transistors sound when they overdrive.