I agree with you that most books do not follow a logical path when defining thermodynamics terms. Even great books such as Fermi's and Pauli's.
The first thing you need to define is the concept of thermodynamic variables.
Thermodynamic variables are macroscopic quantities whose values depend
only on the current state of thermodynamic equilibrium of the system.
By thermodynamic equilibrium we mean that those variables do not change with time. Their values on the equilibrium cannot depend on the process by which the system achieved the equilibrium. Example of thermodynamic variables are: Volume, pressure, surface tension, magnetization... The equilibrium values of these quantities define the thermodynamic state of a system.
When a thermodynamic system is not isolated, its thermodynamic variables can change under influence of the surrounding. We say the system and the surrounding are in thermal contact. When the system is not in thermal contact with the surrounding we say the system is adiabatically isolated. We can define that,
Two bodies are in thermal equilibrium when they - in thermal contact
with each other - have constant thermodynamic variables.
Now we are able to define temperature. From a purely thermodynamic point of view this is done through the Zeroth Law. A detailed explanation can be found in this post. Basically,
We say that two bodies have the same temperature if and only if they
are in thermal equilibrium.
Borrowing the mechanical definition of work one can - by way of experiments - observe that the work needed to achieve a given change in the thermodynamic state of an adiabatically isolated system is always the same. It allows us to define this value as an internal energy change,
$$W=-\Delta U.$$
By removing the adiabatic isolation we notice that the equation above is no longer valid and we correct it by adding a new term,
$$\Delta U=Q-W,$$
so
The heat $Q$ is the energy the system exchange with the surrounding in
a form that is not work.
Notice that I have skipped more basic definitions such as thermodynamic system and isolated system but this can be easily and logically defined in this construction.
You are mistaken. You seem to be assuming that there is some kind of inertia in the process of heat transfer, as in water sloshing about in a tank. There is no such inertia here, so there is no oscillation.
You write:
Since heat has been transferred from A to B, unless I'm mistaken this will place B momentarily at a higher temperature than A.
Yes, you are mistaken about this. The Fourier Model is continuous: a finite excess of heat energy is not transferred which makes B hotter than A. In the model, the amounts transferred are infinitesimal, and cause infinitesimal changes in temperature.
You are also missing that there is a reverse process going on at the same time. As soon as the temperature of B rises even a little, the rate of heat flow back towards A also increases. There is no time delay in waiting for the 'heat wave' to be reflected from the far end of B.
In the Fourier Model, the process of heat flow out of each element of material is not directional. It is a diffusion process which happens at random equally in all directions regardless of the temperature of adjacent elements. But the amount of out-flow increases with temperature, with the result that there is a net flow of thermal energy away from regions with higher temperature and towards those with lower temperature.
When B reaches the same temperature as A there is a dynamic equilibrium between the flows from A to B and B to A. Heat does not continue flowing preferentially from A to B because of inertia. When you apply the mathematical model the result is not an oscillation, but an exponential decay in the temperature difference between A and B.
Outside of the ideal model, because heat flow is a random process, in reality there are small random fluctuations in temperature between two bodies which are in thermal equilibrium. However, these fluctuations are insignificant unless the bodies are microscopic (or your thermometer is highly accurate), and they are not oscillations.
Best Answer
There are two infinite reservoirs, one at $T_1$ and the other at $T_2$, right? So suppose your object's a homogenous rod, one end in contact with the $T_1$ reservoir, the other end in contact with the $T_2$ reservoir. Then, forgetting the ambient temperature part, each end will (must) stay at its respective $T_1,T_2$ temp, just by the problem statement itself. And heat will therefore flow through the rod from the hotter to the colder end. The middle of the rod clearly ends up at $(T_1+T_2)/2$ -- what else could it be? And then if you re-apply that reasoning to $1/4,3/4$ way along the rod, etc, you can see the temp along the rod varies linearly.
Introducing an ambient temp, then there's also a $T_0$, i.e., all surfaces not in contact with the $T_1,T_2$ reservoirs are in contact with the $T_0$ "ambient reservoir", so to speak. And now all these surfaces are held at that $T_0$ temp, and heat's flowing every which way. And dealing with these general kinds of boundary conditions typically involves solving the heat equation, https://en.wikipedia.org/wiki/Heat_equation