Geometry: Bisection of an angle and constructing a right angle

When bisecting an angle, some texts suggest that that the arcs drawn from $D$ and $E$ and intersecting at $C$ should be of the same radius as that of the arc drawn from $O$ but it is not necessary. The arcs from $D$ and $E$ should be of any length greater than $DE$ (Fig 1).

Similarly, in order to construct a right angle at a point on a line, there seem to be two methods- drawing a large arc from the point and keeping the same radius drawing four more arcs to finally meet at $U$, then joining the intersection point $U$ and the point on the line (Fig 2).

Another method of constructing the right angle can be drawing a semicircle from the point on the line and then bisect it (Fig 3).

My point is, are there two different ideas working here where in the first we keep the radii of the arcs same and in the second one radii can be of any length (greater than some minimum length)? If we can bisect an angle simply by taking any radius of the arc and also construct a right angle by simple method of bisecting an $180^\circ$ angle then why there is a long process of constructing two $60^\circ$ angles at the Point $P$ (Fig 2) then bisecting the other $60^\circ$ angle to get a $90^\circ$ angle?

For the first, there is nothing different going on. Draw $DC$ and $EC$. Triangles $ODC$ and $OEC$ are congruent in either case as corresponding sides are the same, so the angles at $O$ are equal.
For the second, either method works. I have never seen the complicated one, but you are correct it constructs a $60^\circ$ angle and bisects it. They both create the same perpendicular. Whether you think bisecting a segment vs. bisecting an angle means "something different is going on" is in the eye of the beholder.