Why is this a problem?
In the case outlined by Andrea Borruso in the post linked above, the problem is that the visuals are unconvincing to local residents if they do not reflect the actual human geography that they are familiar with.
In the case of my colleague Ornaldo, defining where a given town actually is has an impact on the data themselves. You can find more details about his work at this link, but here is his problem in brief. One of the datasets distributed by Copernicus provides detailed data on temperature for the whole period 1961–2018, with data available for a grid covering all of Europe with a precision of 5.5 km. These data are continuous and know no borders, but change of temperature differs considerably even in contiguous grid cells, in particular in mountain or seaside areas: readers familiar with the geography of Italy, for example, will surely understand that a large number of municipalities falls into one of these two categories.
In order to make these data available to the wider public, Ornaldo created an interactive interface enabling readers to find data about their own town (currently available for Italy, forthcoming for the rest of Europe). Here is the problem: when talking about the temperature of Aosta, people will have in mind the city of Aosta. For example, when they look at the weather forecast, they see the expected temperature of downtown Aosta, not the one on the neighbouring mountaintop, which would be very different and not really helpful in understanding if they need a coat or not to go out. As a consequence, when reporting temperature change of Aosta, if we want to be fair to the reader, we should give the temperature change in the city of Aosta, not on some mountain in its proximity. This is true also for small municipalities in the mountains, especially if we publish these data with a local audience in mind.
Ultimately, this is relevant also for national audiences, considering that such mountain locations are often outliers and as such are more likely to make headlines: to the extent that it is possible, such headline should make sense, and not simply record whatever municipality happens to have more mountain pastures and peaks within its administrative boundaries.
The solution: towns are where people are
Ultimately, towns are where people are. So the best way to find the centre of a municipality could well be to find the population-weighted centre. Given that Eurostat distributes a population grid with data on the number of residents in each square kilometer of the continent, it should be possible to consistently apply this approach for all municipalities in the European Union.
How does this look in practice? Let’s start from an easy case: a relatively big town with large non-inhabited areas within its municipal border. If the town centre is where people are, and through the population grid we know where people live, we can just make a weighted average of the coordinates of the centroids of each cell of the population grid. Et voilà, we have a point that can reasonably be understood as a central location of the town.