I don't like these types of questions, because they are ultimately philosophy, not physics, but I'll give it a shot. First, I don't care so much about the nitpicky distinctions you make between "instants" and "durations", since you can go from one to the other by subtraction. But the distinction you are getting at between the observational and direct "clock reading" concept and the abstract "coordinate time" is philosophically significant, although physicists don't care to clarify it too often. After writing this verbose empty answer, I understand why they don't.
Like all other philosophical questions, one must take the point of view of operationalism and, more specifically, logical positivism. The notion of "instants", "duration", and "time" are only defined to the extent that these have corresponding sense-impressions that we can feel or have instruments which sense them, and ultimately, the goal of a description is to give all the relationships in nature between these sense-impressions, in as precise and predictive a mathematical language as possible. When you do this, you have a formal description of nature.
The description in physics is through a mathematical model with it's own set of ingredients which are abstract computational ideas, things you would put in a computer simulation of the events in the thing itself. Sometimes this model includes an abstract notion of time, as in Newtonian mechanics, special relativity, or general relativity, and sometimes the notion is either approximate or emergent, like when you try to talk about bulk-time in the most fundamental versions of string theory. When you are talking about a notion like coordinate time, you are talking inside the model, about the model notion of time.
The principle of logical positivism or operationalism doesn't quite say that the model should only involve primitive notions directly related to sense impressions, it rather says more subtly that two descriptions that agree regarding all the results of all the measurements and sense-impressions are to be identified, they are fundamentally the same. So if you ask about electromagnetism "Is the world really in axial gauge or Lorentz gauge?", you are asking something ridiculous--- you pick a certain gauge to calculate the answer and give meaning to the vector potential, but the end result doesn't care about the gauge, so the question is meaningless metaphysics. When you introduce extra ingredients beyond the direct observations in a model, you don't discriminate between two versions that are observationally equivalent.
When you have a model which is either exact or exact enough, and a map between this model and reality, when you see a system in the world, you need to identify the class of mathematical representations which correspond to this system by a series of measurements and by your knowledge of the correspondence between the world and the model, and then the model tells you the best predictions.
When the concept "coordinate time" occurs in the model, the way in which we map the abstract concept to clock time, the reading of the clock, is ideally though a good model of the clock, given in terms of coordinate time. So that the clock instants are representing the coordinate time correctly when the model of the clock predicts that the separation between the clock readings are uniform in the abstract model.
You might object that this is a circular definition (this is why I hate this philosophical stuff--- one gets annoyed when one produces such a big pile of words to clarify something so obvious) --- you need to introduce the notion of coordinate time and models of object inside coordinate time in order to even define what it means to measure coordinate time, so the notion of coordinate time might not look exactly operational, because verifying that the operational definition works relies on having a model which uses the very same notion underneath. But this is not really a circularity--- you assume the model works, you verify that the measurement works in the model to measure the abstract time, then you run the clock, and it works as it should. I mean, if it doesn't something has gone wrong in the model, you discovered that the laws you have aren't right.
The notion of coordinate time, at least to the extent one can measure it by clocks, is ok as far as logical positivism is concerned, because you only consider it to be meaningful to the extent that you can measure it, and you don't distinguish between two models with identical predictions. For example, people on this site often ask "What would happen if time starts running twice as fast, all of a sudden?". The answer is nothing, it is not meaningful to ask this without an operational definition of what exactly is different. This can be asked about space too: "What would happen if all atoms suddenly became twice as big and twice as far apart?". This question is often asked in this form, which is identical, but looks less silly: "What would happen if the speed of light suddenly changed?" All of these are operationally meaningless--- you need to say what these dimensional things are changing relative to, you need an operational definition. So you can ask "What would happen if the wavelength of light emitted by Hydrogen in the 2P-1S transition suddenly became twice as short compared to the wavelength of light emitted by electron positron annihilation?", or you can ask "What would happen if the mass of the proton were twice as much in Planck masses?" but you can't ask "what would happen if the speed of light was variable?"
To summarize this godawful nonsense, in physics you make a hypothesis regarding the ingredients of the mathematical model, in this case that it has a coordinate time, and you then model the world. If at the end of your analysis, you find that a pendulum ticks in equal amounts of coordinate time, and you are confident that this model corresponds to real pendulums, then you say "this clock measures coordinate time", even though the pendulum is measuring pendulum clock time, which technically is only related to coordinate time to the extent that you have a better operational definition of coordinate time, better than the pendulum.
You are allowed to use the concept of coordinate time freely only to the extent that you can imagine a sequence of better and better clocks, constructed more carefully, and idealize the notion of coordinate time in the limit.
This is the assumption that breaks down in quantum gravity, you can't measure durations more accurately than a Planck time, and in this case, you should never say "this clock measures quantum gravitational coordinate time exactly", because the concept doesn't make operational sense, and it shouldn't (and doesn't) appear in a correct quantum gravitational theory. What you should say here is something more subtle, that an asymptotic scattering experiment tells you the quantum directions and phases of the outgoing particles for the incoming particles, where the incoming particles make an approximate clock that lasts for a while, and the outgoing particles are photons shining on it's face to tell you what it reads. There will always be uncertainty in the actual time defined operationally this way, since the space-time description is not accurate, but the scattering matrix in flat space survives the transition to quantum gravity, and this concept can replace the older concept.
For ordinary non quantum-gravitational physics, the concept of time is ok, since you are far from where it's going to break down. So you can see from the agreement of pendulum and spring clocks that there is a correct notion of macroscopic time that these are approximating, for scales much larger than those of quantum gravity. I am ashamed for having writing so much, because I think this question, outside of quantum gravity, is rather self-evident to anyone who doesn't worry about the exact words one associates to a philosophical position. It should have a one-word answer, like "conceptmodelization", but if this word exists, I don't know it.