Answer
Basically there are three different definitions assigned to the term angle as explained here. I am not familiar with the notations used in the link that I've given. So I cannot comment on it.
Your question is completely legitimate because in some circumstances the angle is taken as 0 degree if something is been rotated one complete circle(e.g. in case when we are talking of angle between vectors.)
When we use the term angle in the definition of Angular displacement$^{[1]}$ we are infact referring to the term rotated angle for the term angle. So in the situation that you have depicted in your question the definition of an angle as given in the established-text$^{[2]}$ is applicable for the term angle.
From the definition of an angle given in the referred established text it is clear that if something is rotating about a point and it covers a complete circle, we should take its angular displacement as 360 degree not 0 degree.
Your main question is:
"If something is rotating about a point and it covers a complete circle, should we take its angular displacement as 360 degree or 0?"
In short the answer is:
We should take its angular displacement as 360 degree not 0 degree because of its(angular displacement's) definition.
You also say "Is angular displacement ambiguous?"
No, suppose something covers one rotation in anticlockwise direction then we say it is rotated through an angle $+2\pi$ but if it is suddenly rotated back in clockwise direction we say it has covered an angle $-2\pi$ now so the net angle it covers is $0$. this is what displacement is all about i.e net change in parameters which is now $\theta$.
${}^{[1]}$Definition of Angular displacement:
Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis.
References
A Paragraph written in the Book: Fundamentals of Physics (8th Edition)
"we do not reset $\theta$ to zero with each compelete rotation of reference line about the rotation axis. if the reference line completes two rotations from the zero angular position then the angular position of the line is $\theta = 4\pi$ rad."
Established text
"Unified algebra and trigonometry (Addison-Wesley mathematics series) article 3-5"
${}^{[2]}$3-5 Angles.
In geometry an angle has usually been defined as the configuration consisting of two half-lines (rays) radiating from a point. However in trigonometry we generalize the definition by stating that an angle thus defined by two half lines has a measure which corresponds to the amount of rotation required to move a ray from the position of one of these lines to the other. consider the figure$^{[3]}$ with two lines $m$ and $n$ intersecting at $o$ and lying in a plane prependicular to our line of vision. if we consider $m$ as initial line and $n$ as terminal side of the angle $o$ as its vertex, there are two possible directions of rotation of the initial side $m$. The angle is said to be positive if the rotation is counter clockwise, but negative if clockwise. A curved arrow will indicate the direction of rotation.
let us now consider a ray m which issues from the origin of a rectangular coordinate system and coincide with the positive x-axis(Fig 3-9). As this ray rotates, any point $P$ on $m$ will trace out part or all of the circumference of circle of radius $OP$. In fact the circumference may be traced several times.
After the rotation $OP$ will be in some position $OP'$, where the circular arc $\stackrel \frown {PP}^{'}$ denoted by $s$, may be used to measure the $POP^{'}$ is said to be standard position, and to be in quardrant in which its terminal side $OP^{'}$ is located .
The most logical units for measuring the magnitude of an angle $POP^{'}$. An angle would seem to be the number of revolutions due to the rotation from the initial to the terminal side of the angle. since the number of revolutions of any angle is determined by the ratio of the intercepted circular arc length $s$ to circumference of the circle we define, magnitude of an angle in revolutions as
Angle in revolutions$ = \frac{s}{2\pi r}$
For example if $P$ traces out an arc half the circumference, the corresponding angle is one of one-half a revolution. Likewise if arc is twice the circumference, the angle measure is two revolutions.
Consider the two coencentric circles at $o$, in Fig.3-10, with $\stackrel \frown {PP}^{'}$ as arc of length $s$ on the circle of radius $r$, and $QQ^{'}$ an arc of length $s^{'}$ on the circle of radius $r{'}$. By using the theorem that similar triangles have propotional sides, and recalling the definition of arc length from article 3.4 it can be proved that
$$\frac{s^{'}}{r^{'}} = \frac{s}{r},$$ and therefore,
$$\frac{s^{'}}{2\pi r^{'}} = \frac{s}{2\pi r}$$
The magnitude of any angle is thus independent of the length of its initial or terminal side. Although the use of revolutions is the most natural method for measuring angles, there are other more convenient systems.
The system most commonly used in elementary work such as survyeing and nevigation is sexagesimal, in which the degree is the fundamental unit. In this system one revolution = $360^0$, $1^0 = 60^0$(minutes) and $1^{'} = 60^{''}$(seconds)
Angle in degrees = (revolutions) $360^0$
For example, one-half a revolution is $180^0$, or an angle of two revolutions is $720^0$.
The other important system used in ... ...
Angle in radians = (revolution) $2\pi$
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${}^{[3]}$I could not add the figure. Please refer to the original context.
