Were there other related definitions defined around this time?
In physics textbooks, vectors are introduced first, and then vector fields; given the complexity of the notion of tensors, one would suppose tensors would then be introduced and then tensor fields. Instead, a quite complex and obscure definition of a tensor field (and not a tensor) is presented without enough motivation to the curious - this is the definition you have more or less presented.
That the definition is obscure has been understood, with various physicists, such as Dirac, in his short book on GR, noticing the explanatory lacuna and who suggests an 'affine' description of a tensor would be worth looking at before moving to the notion of a tensor in general; but he doesn't attempt a description of an 'affine' tensor. Weinberg, in his book Gravitation and Cosmology, does make an attempt at it but the result is not much more clear than the original definition.
Moreover, one would suppose that Einstein himself, given the importance of tensors in his theory of gravity, would make an attempt to explain the notion in his book with Infeld, The Evolution of Physics; but whilst they pays careful attention to the notion of a vector - they, unfortunately, they do not discuss the notion of a tensor. This is simply, as I've already noted, that the actual definition used by physicists, isn't intuitive (in fact, it is, when thought through but nevertheless, it is long-winded and relies on an operational definition rather than an intrinsic definition).
The main alternative presentation of tensors is most often used by mathematicians rather than physicists, however, this tends to be increasingly not the case with more advanced work. In the future, given the importance of tensors, I expect this to filter down to college or even high-school, as the notion is not at all complex, and is in fact intuitive.
It's presented, for example, in MacLanes book, Categories for the Working Mathematician (first published in 1971) by what is known as universal properties and he notes, that although Bourbaki, in principle, came close to this definition, but given that they didn't have the notion of a universal property to hand - they missed it.
It's obscure enough that two moderators (hi @danu, @HDE226868) deleted my original post complaining that I had been mistaken in what I had written and that I had only presented an alternating contravariant tensor 2-field!!
Not so. I had sketched out the definition of a 2-tensor and not a 2-tensor field - nor was it contravariant or covariant - this relies on the imposition of a basis but since the definition is basis free, we don't need this notion although it is consistent with it.
To properly introduce tensors in this kind of generality we would need to introduce the apparatus of a manifold which is already extra information. We can dispense with this in the same way we can dispense with the notion of a manifold in discussing a vector tangent to a point of a manifold and simply discuss the simple and straight-forward notion of a vector.
Now, vectors can be introduced algebraically, geometrically and physically. Likewise this can be done for tensors.
Geometrically speaking, tensors are generalisations of vectors where instead of generalising the space the vector lives in, we generalise the nature of the arrow itself. Thus a 2-tensor has two magnitudes, and two directions (the edge elements); it's basically a directed area element together with an actual area magnitude (that is bilinear on the edge elements). Unlike vectors, given that it has two directions, there is a notion of internal scaling. For example, we can rescale one edge by $k$ whilst rescaling the other edge by $1/k$; this preserves the area, but not of course the area element.
The most precise definition, and the clearest definition - to the initiated - is the description by universal properties; you can find it in McLanes book, Categories for the Working Mathematician; it is constituted by a bilinear map
$\otimes: U \times U \rightarrow U \otimes U$
this is the tensor map; it associates the 2-directed area element from a choice of two directed edges, say $u,v \in U$
But how is this defined; this is the clever bit when it comes to universal properties - which in a sense is misnamed - it really out to be called a characterising property as it characterises the property up to isomorphism.
Given a directed area element, like a rectangle, we are not just interested in the rectangle we also want to assign it an actual area; now, when we measure the area of a rectangle we have actually two different ways of doing this:
We can measure the lengths of the edges and multiply the two together. This is the area determined by the edges; and it is bilinear on the edges
We can actually measure the rectangle itself, perhaps by paving it with smaller rectangles; this is measuring the rectangle as a rectangle. This is not bilinear, but linear.
The two should obviously match.
The universal or characterising property says that for any choice of area map determined by the edges there is a unique area map determined by the rectangle; in fact, to be precise, we also have to allow vector valued areas (note that this generalises the notion of an area, as typically areas are measured by the ground field and this is obviously a vector space).
Thus to complete the definition we say, for any other vector space $V$; then given any bilinear area map $A:U \times U \rightarrow V$ then there exists a unique $A':U \otimes U \rightarrow V$ such that $A'\otimes(u,v)=A'(u\otimes v)=A(u,v)$.
Now, after choosing a basis we get the transformation property beloved by physicists; moreover, when we lift this to a vector bundle over a manifold we get a contravariant 2-tensor field; when we specialise this bundle to the tangent bundle, we get a contravariant tangent 2-tensor field; to get the covariant twin, we merely specialise it to the cotangent bundle; if we then choose a basis, we more or less get the definition that you have presented.
How has today's definition of a tensor changed compared to its beginnings?
There is a lot more to say; for example, the notion of a tensor has been abstracted into what was originally called a tensor category; but is now called a monoidal category - these are important in characterising TQFTs (topological quantum field theories) of which the most important example, physically speaking, was the exact quantisation of the vacua of 3d gravity.