You effectively answered your own question, but I'll add some evidence.
First of all, these are the clothes Halloway was wearing when he was turned invisible. Of particular note are the white shirt, the dark suit jacket and trousers, and the dark shoes, because we see these again and again in later scenes. (The waistcoat and tie are discarded in his apartment.)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/sROCP.jpg)
Shortly after the accident that turned him invisible, Halloway mentions that he can't see his body or his clothes, indicating that the clothes he was wearing at the time were turned invisible, just as large portions of the building he was in were.
I walked over to the mirror. I was right. There was no reflection. My body, clothes, everything was gone. I was invisible.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/KdF40.png)
However, he puts on a hat that was hanging on a coat stand in a portion of the building that wasn't turned invisible, and that's perfectly visible, to him and everyone else.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Tg2iN.png)
So the rules are clearly established here. The clothes he had on when he was turned invisible are also invisible, due to the same accident. But any clothes which weren't affected by the accident -- such as a hat located elsewhere in the same building -- are visible, and remain visible even if he wears them. These rules apply consistently throughout the film.
Note that when Halloway flees from his apartment the following morning to avoid being captured by Jenkins' goons -- as shown in the second of the two videos you posted in your question -- he puts on the same clothes he had on when he was turned invisible. They look the same (white shirt, dark suit jacket and trousers, dark shoes), and you can see him feeling around for the jacket with his hands, and fumbling while trying to put it on, because he can't actually see it.
Halloway was wearing visible clothes when Jenkins' goons showed up again at the beach house, because he thought they wouldn't find him there, but wisely changes back into the invisible clothes before escaping in a car with Alice, to reduce the chances of being seen and caught.
To summarise, in any scene where Halloway wants to be fully invisible -- like when he's fleeing or hiding -- he either strips naked, or wears the same clothes he had on during the accident (usually the latter). He only wears different clothes when he wants to be seen, or when he's alone, and those clothes are always shown to be visible.
The first of the two videos you posted is one such example, but there are more.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/rsxua.png)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/hjRNZ.jpg)
In the scene below, Halloway strips fully naked to avoid being seen, when his friend, George, unexpectedly turns up with guests at the beach house he was hiding out in. Again, note that the clothes he took off in this scene (the ones George had just picked up off the floor in the image below) are not the same clothes he was wearing during the accident.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/YABTL.jpg)