Eliza, I am going to make one more attempt at answering your question; I'm unable to modify my earlier answer, for some reason, so I will just have to rewrite it.
To begin with, Temperature, and the practical measurement or control of Temperature, are thermodynamic properties of MACRO systems, and have nothing whatsoever to do with quantum mechanics (QM), despite the childish efforts of the trolls, to introduce qm into the discussion.
The sensing and control of Temperature is a major issue of modern science (physics, chemistry etc) and technology. For example, the semiconductor industry is dependent on controlling Temperatures, up to several thousand K, often to small fractions of a degree.
So your system of water, plus a mercury thermometer, to "measure" its Temperature, actually contains a lot more than you describe, and each of those elements will introduce uncertainty into the system. You mention waiting for heat to flow from the water to the thermometer, and reach equilibrium. That implies that your Temperature is stationary, or is supposed to be; otherwise it isn't at equilibrium.
But your container of water, is also connected by thermal paths, to various ambients, that each have their own Temperature; higher or lower than your water. Most likely, is that you have some "heater" that is supplying energy to the water container (heater element), or to the water (microwave), and each of these factors, including the thermal resistance of the paths, is a variable with some uncertainty. The water and container, also lose heat by multiple paths to external environments at other Temperatures, which themselves are uncertain.
Your thermometer, as well, in addition to having a thermal resistance connection to your water, has other thermal paths, and resistances, to the environment, and maybe, even to the external heat source. Again you have variables and uncertainties, involved in a rather complex thermal equivalent circuit.
Ultimately, you are going to get a mercury liquid level, designating THE Temperature reading output of that thermometer. Or it could be the output of some quartz crystal thermometer, or Platinum resistance element.
But you are wanting to know the Temperature of the water, and there is no certainty, that the reading on the thermometer (any kind), is actually the Temperature of your water.
If you were trying to grow a crystal, or thermally diffuse some impurity dopant into your material; or even bake a cake, or cook some food, that material itself, is only concerned with the Temperature, that IT perceives. It cares not about your thermometer, or what it reads.
This uncertainty, is often called "instrumentation error", and it is one of the most common problems of process control. It can get much worse too. Some people think they can instead measure some proxy stand-in for their Temperature, and simply ASSUME that there is a robust transfer function between the actual variable (Temperature), and the proxy substitute (maybe a color, or a mechanical motion).
Well the further you get from OBSERVING the variable you wish to control, the more uncertainty, there is in the robustness of your measurement.
Your pot of water sitting on its heating element, had a certain range of Temperatures, throughout the water; but when you insert the thermometer into the system, you changed the system, and in ways almost impossible to accurately quantify, so you have introduced an uncertainty into the new system, and that requires good experimental skills, to keep down to acceptable levels.
All observations and measurements in physical systems, are altered just by providing the sensing apparatus, and there is no way to eliminate that. Yes, it can be minimized, but never eliminated.