I don't think your implications are the right way to think about this kind of operational definition. Rather, we should consider this as the way physicists talk about the local data attached to various geometric objects over spacetime.
In differential geometry, we have the general notion of a fiber bundle $\pi : B\to M$ over a manifold $M$. Almost all objects we usually talk about live "in" such bundles - they're sections of them or structures on them.
For such bundles, we have that $\pi^{-1}(x)\cong F$ for whatever the fiber $F$ is - a vector space, a space of tensors, etc. While globally $B$ carries generally more structure than just attaching a copy of $F$ to every point in $M$, locally there exist trivializations of these bundles: A cover of $M$ by open sets $U_i\subset M$ for which $\pi^{-1}(U_i) \cong U_i\times F$, i.e. if you only look at one of the $U_i$, the bundle is just attaching a copy of $F$ to every point in $U_i$ in the straightforward way.
Those trivializations come with transition functions $t_{ij} : U_i\cap U_j \to \mathrm{GL}(F)$, where by $\mathrm{GL}$ I mean the appropriate set of invertible structure-preserving transformations of the fiber, e.g. the linear invertible transformations of a vector space when $F$ is a vector space. These functions carry the information about the global structure of $M$ and $B$ - they tell us how to glue together $B$ again from this local data: For any $x\in U_i\cap U_j$, we should consider $(x,f)\in U_i\times F$ and $(x,t_{ij}(x)f)\in U_j\times F$ to be the same thing.
This procedure can be reversed (and is sometimes known as the "cocycle construction" of bundles): We can start with a bunch of $U_i$, attach our desired fiber $F$ to them, specify a bunch of $t_{ij}$, then define $B$ to be the disjoint union $\bigsqcup_i U_i\times F$ quotiented by the relation $$(x_i,f_i)\sim (x_j, f_j) \iff x_i = x_j \quad \land \quad f_i = t_{ij}(x_j)f_j.$$
Now I claim that the specification of the $t_{ij}$ is equivalent to what the physicists are doing when they define objects by their transformation behaviour: When we say that "a vector" $v^\mu$ transforms like $v^\mu \mapsto J^\mu_\nu v^\nu$ under "coordinate transformations", where $J$ is the Jacobian of the transformation, in this language what we mean is that we're building a vector bundle with fiber $\mathbb{R}^n$ - the tuples of numbers $v^\mu$ - by specifying that when $U_i$ and $U_j$ are two different coordinate charts, and $\phi_{ij} : U_i\cap U_j \to U_i \cap U_j$ is the coordinate transformation between them, then the transition function between $U_i\times \mathbb{R}^n$ and $U_j\times \mathbb{R}^n$ is given by the Jacobian of $\phi_{ij}$, $J(\phi_{ij}) : U_i\cap U_j \mapsto \mathrm{GL}(\mathbb{R}^n)$.
This generalizes straightforwardly to tensors (building bundles with fibers $\mathbb{R}^n\otimes\dots\otimes \mathbb{R}^n\cong \mathbb{R}^{n^m}$), gauge fields/connections (fiber is the Lie algebra of the gauge group, the transition functions are the behaviour under local gauge transformations), etc.
Statements like "the difference of two connections is a tensor" just means examining the way the difference of two connections is acted on by the transition functions - it turns out the "non-tensorial" part of the gauge transformation drops out of the difference, since it is independent of the specific value of the connections, and so this lives naturally in the bundle we defined with the tensorial transition functions.
