A common definition of a scalar Some dictionaries define a scalar as follows:

A quantity, such as mass, length, or speed,  that is completely specified by its magnitude and has no direction. -- The Free Dictionary

However, it is my impression that in many contexts scalars can be signed, in which case their magnitude (their absolute value) does not specify its value.  This definition is even used on a test question here.  Is it true that this definition is inaccurate?
 A: From a geometric object perspective, a scalar is a rank 0 tensor and, as such, is invariant under rotations of the coordinate system.
The tensor contraction of, e.g., a one-form and a vector is a scalar, i.e., a real number.
A: You must always say with respect to what something is a scalar.
If we are given a group $G$, something is called a scalar if it is a member of the trivial representation of that group, i.e. if the (symmetry) group does nothing to it. Nothing more, nothing less.
In the most common situation, this means that a scalar is a scalar under the rotation group $\mathrm{SO}(3)$, and thus simply a real number instead of a vector or some matrix/tensor.
There are also pseudoscalars, which are scalars w.r.t. $\mathrm{SO}(3)$, but not w.r.t. to the full orthogonal group $\mathrm{O}(3)$.
A: The dictionary definition is wrong. For example, time is a scalar in Newtonian mechanics, and time can be negative. That means that time is not completely specified by its magnitude (absolute value). Other examples include charge, energy, and Celsius temperature.
The definition could be improved by cutting "is completely specified by its magnitude" and clarifying "direction" to be "direction in space." We'd then have this definition: a scalar is something that has no direction in space, i.e., if you rotate it, it doesn't change.
A: You're simply reading too much into the word "magnitude." You want to translate it into technical terminology as "absolute magnitude," since the latter is often abbreviated to "magnitude" by physicists anyway.
But in everyday parlance "magnitude" is only trying to convey comparability. You have two things, each with their own magnitude, and the implication is that the magnitude of the one is equal to, greater than, or less than that of the other. This ordering property is exemplified by the real numbers. So "magnitude" → "real number" is better in certain contexts than "magnitude" → "absolute magnitude."
Your dictionary is entirely consistent (and consistent with the way I speak in the English vernacular). Note that for "magnitude" it gives

A number assigned to a quantity so that it may be compared with other quantities.

and

A property that can be described by a real number [...]

