Ultimately, you make a totally legitimate point; we very well could have defined the term "four-vector" to refer to a type of object that transforms in a different way, but we make the particular definition that we do because it is useful to have a term that refers things that transform like spacetime positions when you change frame. Here are two reasons why:
Fact 1. Given any two four vectors $A^\mu$ and $B^\nu$, the quantity $g_{\mu\nu}A^\mu B^\nu$ is invariant under a change of frame.
Notice that this would not have been true unless $A^\mu$ and $B^\nu$ were four-vectors because the proof of this fact relies on the metric being preserved by Lorentz transformations, and not by other arbitrary things. Here is another reason why the definition is useful
Fact 2. Lots of really useful and physically significant quantities happen to be four-vectors. Take, for example, $J^\mu$ and $A^\mu$ (the current and vector potential) in electromagnetism.
Having said all of this, however, note that there are tons of other quantities that do not transform as four-vectors when one changes frame. In fact, given any representation $\rho$ of the Lorentz group, one often encounters quantities $Q$ that transform as
$$
Q' = \rho(\Lambda) Q
$$
For example, there are objects called Weyl spinors that transform as
$$
\psi' = \rho_\mathrm{weyl}(\Lambda)\psi
$$
when one transforms between frames.
The upshot of all of this is the following
Upshot. Lorentz 4-vectors are not special. However, since every change of reference frame can be associated with a Lorentz transformation, every quantity that you want to transform between frames must necessarily transform in a way the depends, in some way or another, on the Lorentz transformation between the frames. This leads us to not only define four-vectors, but a host of other objects that have specified transformation laws under changes of frame and to give them special names. Doing this is useful because such objects appear all over the place in physics, and we can prove useful properties about objects with certain transformation behaviors.
Therefore it occurred to me that the definition in the book of Weinberg is not consistent with that in the book of Tung: in one of them the symbol ${\Lambda_\mu}^\nu$ is defined as the inverse of the Lorentz transformation of contravariant vectors, while in the other case, the same symbol is defined as the transpose of the original matrix.
The symbol $\Lambda_\mu{}^\nu$ is not defined to be the transpose of the original matrix. The transpose of the original matrix is ${\Lambda^T}_\nu{}^\mu$ (assuming that the original matrix is $\Lambda^\mu{}_\nu$). You have to keep the "$^T$". So long as you use "$^T$" to tell the difference between the matrix and its transpose, everything should work out with no inconsistencies.
Best Answer
Indeed, to add to orion's answer, the definition of a vector (and / or spinor) in physics (as opposed to the mathematical definition as an element of a linear space over a field) is most often stated in terms of how the object concerned transforms (e.g. see my answer here) when "co-ordinate transformations" are made: more precisely, when one switches between two overlapping charts of a manifold. Vectors transform like the members of tangent spaces to manifolds. "Covectors" or one-forms are linear functionals of vectors. Then tensors are simply general multilinear functionals of vectors and / or one-forms. I really like the language and teaching style of Misner Thorne and Wheeler "Gravitation" here, also replicated in the early part of Kip Thorne's lectures here.
The Lorentz transformation, on the other hand is a kind of co-ordinate transformation, and, as such, a vector / oneform / tensor must, by definition, transform in the prescribed way by it.
So tensors, vectors and n-forms are defined by how their components behave in response to co-ordinate transformations. If you like, tensors, vectors and n-forms are a kind of "software" (or "machine" to use Kip Thorne's and Misner/Thorne/Wheeler wording) that is built to a specification of how that software/machine must react to various inputs. In this analogy, the Lorentz transformation is one of the inputs for the machine addressed by the specification.