Is there an agreed upon physics definition of the term 'speed'?--for example, can it be negative? The term speed is commonly defined as follows:
https://www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors
Speed, being a scalar quantity, is the rate at which an object covers distance. The average speed is the distance (a scalar quantity) per time ratio. Speed is ignorant of direction. On the other hand, velocity is a vector quantity; it is direction-aware. Velocity is the rate at which the position changes.
Then see:
http://www.statisticshowto.com/scalar-definition/
Can a Scalar be Negative?
A number like -10 can be a scalar or a vector depending on what situation you are using it in. In linear algebra, scalars can be negative. A negative scalar like -10 would result in a vector in the opposite direction. In physics, scalars and vectors are defined by what happens to them during rotations. Direction is sometimes denoted with a + or – to mean the positive or negative direction relative to a reference point. In this situation, -10 would not be a vector as -10 would mean 10 units in the negative direction from the reference point. To avoid confusion, the word “scalar” in physics is sometimes limited to complex numbers. 
For an example of confusion see:
https://www.wired.com/2014/06/whats-the-difference-between-speed-and-velocity/
Also 'speed' is also often defined as the magnitude of the velocity vector, and it seems like this can lead to confusion.
See
Does a resultant magnitude always have to be positive?
My question is:
Is there an agreed upon physics definition of the term 'speed'? For example, can the speed variable be negative or does it have to be preceded by a negative sign?
 A: I have always heard and used speed to mean the magnitude of velocity. 
Average speed would then be the average of instantaneous speed. 
Note that this might be different than the magnitude of the average velocity (i.e. the average of the instantaneous velocity).
The magnitude of something (real number, complex number, real/complex vector) is always a positive real number (in the usual usage of the term magnitude).
Scalars are real or complex numbers. Recall that real numbers are a subset of complex numbers.
Since magnitudes are real numbers they are always scalars, albeit they are real positive scalars, a particular subset.
I don't see any potential ambiguity in any of these points.
Sometimes when we treat 1-D kinematics problems we can forget the fact that displacement and velocity are vectors and just assign numbers to them. If we want to be really pedantic we can note that the real numbers are in fact a 1-D vector space, so these numbers we are assigning can in fact be thought of as vectors. But this is not important. We just have to note that it is possible to assign a velocity of -10 m/s to an object and note that since the magnitude of $-10$ is equal to 10 then the speed of this object would be 10 m/s.
$$
|-10| = 10
$$
To summarize and answer your last sentence: The speed variable cannot be negative. If you want to write the velocity as a function of speed then you must include a magntidue (the speed) and a direction. In the 1-D case this direction is supplied by a plus or minus sign and high dimensional cases this direction is supplied by a unit vector in that space.
edit: See Speed Wikipedia page. Also for completeness I note that for a vector velocity we have
$$
S = |\mathbf{v}| = \sqrt{v_x^2+v_y^2+v_z^2} >0
$$
This holds because $v_i^2$ must be positive and we always take the positive root. in the 1-d case we have
$$
S = |v| = \sqrt{v^2} >0
$$
A: Velocity is generally understood as a vector $v$ and speed as it's absolute value $|v|$, this means that it's always a positive value (where we think of zero being positive too).
If you take a component of velocity in a system of axes then this scalar value can take both positive and negative values - but it's not speed as such.
edit
One possibility where we can have velocity reduced to speed - more or less - is if we look at a 1d system, for example a particle moving along a straight line (actually a curve is possible but a straight line is simplest). Then velocity here is more or less speed, except because it is still a vector (but also still a scalar) it can take negative values!
This example, by the way, shows the importance of lifting up in dimensions to see what a particular physical quantity actually is. For example, if we stayed with the 1d example, we would suppose velocity to be a scalar, whereas when we lift in dimension, we see it is actually a vector.
A: *

*A scalar is a one-dimensional value.

So, every signed scalar (quantity or state) is technically a one-dimensional vector.
Note that not every scalar is a magnitude, which—by definition—is non-negative.


*An Euclidean vector is a value with a free direction in space.

The word free is significant: in $\mathbb R^3,$ the directed distance $-3.7$ units along a given curve is not a vector.
The velocity $-3$ m/s is a vector in 1-dimensional space and a scalar—though not a magnitude/speed—while the velocity $\small\begin{pmatrix} -1 \\ 2 \\ -2\end{pmatrix}$m/s is a vector in 3-dimensional space.

The cited wired.com article is actually quite sound—until the author erroneously infers from (the definition) “instantaneous speed = magnitude of instantaneous velocity” that “average speed = magnitude of average velocity”:

*

*average speed (over a fixed time interval) is proportional to the total distance travelled,

*whereas magnitude of average velocity (over a fixed time interval) is proportional to the shortest distance between the initial and final positions,

so the former often exceeds the latter.
A: Speed is distance traveled per unit time. 
Since distance traveled can never be negative, therefore speed is always positive. Average speed is NOT the magnitude of average velocity. Instantaneous Speed is the magnitude of Instantaneous Velocity. At a very very small distance covered, $dx$, the turn of a body can be neglected and it can be assumed that it follows a straight line path.
Speed is always positive. 
A: 
can speed be negative?  

One way of answering this question is to trawl the literature and use the Google search engine to help with the process.
In particular when looking at images of speed against time graphs negative speeds were not seen to be plotted.  
My conclusion is that most authors regard speed as the magnitude of the velocity and I thought that the following pair of graphs sums this up quite nicely.  
 
I thank @jgerber and @PhilipWood for pointing me in the right direction.
