If the Sun was 2 times closer to Earth, would it have the same...?

Yes, you are correct (mostly). For practical use, distant objects will appear about twice the diameter if you cut the distance in half.

The trouble is that the Sun is not a disk; it is a sphere. Technically the formula is:

Angular diameter = 2 arctan (1/2 d/D) where d is the visual diameter and D is the distance to the object (expressed in the same units)

A tiny amount of the Sun that is visible from a distance would not be visible from the closer view.

From a view twice as close, the Sun's angular diameter would be 1.055 degrees instead of the present 0.53 degrees.

.

If the Sun was ten percent closer to the Earth than what it is now, the Earth would be a lifeless desert, if the Sun was ten percent further away from the Earth than what it s now, The Earth would be an ice desert with glaciers on every corner, so the Earth now is in the habitable zone of the Sun, which is why life exists on the Earth and liquid water exists on the surface.

It would be bigger and you woldnt see it because we would all be dead