Modern Vespa

As the title says I am perplexed and don't know how to fix it. I tried replacing the bulb but to no avail.

The rear light only turns on while braking, but it is off while running

Are there 2 separate bulbs within the light unit itself? Or is it one single bulb that changes brightness when the brakes are pressed?

I'd start at the cheapest option first, which is buy another rear bulb on the off chance that the new bulb has been dropped at some point.

After that, I'd buy a spray can of "contact cleaner". Unplug the connectors near the tail light, spray some in and then reconnect.

Contact cleaner is a mild acid, so try and avoid getting any on the Vespa, and wash your hands immediately after - even if it feels ok in the moment, as it will irritate / burn if left.

I did try the first option, there was only 1 bulb to replace

It sounds like a double filament bulb. They usually go in and a quarter turn to the right. Try not to touch the bulb with your bare hands a clean cloth comes in handy.

this is the exact video that i followed, and the result was what i posted about: the light only turns on when braking, but not while its running

Wrong bulb perhaps, or the socket it plugs into has an issue.

Start by using a simple 12v tester to see if there is current to the bulb at the socket. Work from there.

Isn't the bulb a single filament brake light only? The running light is separate?

This video shows a single filament bulb. It wouldn't have a rear tail light in the same bulb.
Aren't the running lights separate from the brake light on a 2016 Sprint? I would compare it with my Primavera but it's tucked away for the winter.

Yeah. I think the running light is built in to the light housing.

OP there should be another wire coming in for that.

Is it incandescent and diaphanous?

Tailight bulbs are single filament on my 2018 Primavera and 2015 Sprint 150
both RT & LT rear running lights stay on but taillights only come on when braking, pretty sure this is how they are from factory.

I'd also add first video rear running lamps are clear, both bikes of mine are red and come on when the scooter is on

I think that's because of the difference between Euro and US turn signals. The OP hasn't said where they are located.

Possibly mine may very well have been altered before I got them? The front running lamps are clear LED looking. Here is the picture of the rear end, 2018 Primavera 150

In the nursing field there is a joke. When too many people are trying to achieve a simple task they ask how many nurses does it take to change a lightbulb? You're not supposed to answer that.