Choice of baselines for climate modelling

The definition of ”pre-industrial” period, generally used as a reference to discuss changes in global temperature seems to lack consensus. While Global Warming of 1.5°C (IPCC) takes the period 1850-1900 as reference to define “pre-industrial”, some studies like Estimating Changes in Global Temperature since the Preindustrial Period suggest that 1720–1800 is the better choice to set the reference when discussing global temperature limits. In What baseline should we use in our Paris target global warming analyses? the author describes pros and cons of using different baselines. To the risk of completely missing the points outlined in the resources linked: what are the consequences of setting an arbitrary baseline for climate models (e.g. use the 1850-1900 period as used by the IPCC as the conventional baseline)? If we can study relevant paleoecological data points (especially if we can identify relevant data that exhibits similar patterns as some of the observable early consequences of anthropogenic climate change) coupled with historical literature (e.g. see A chronology of global air quality which refers to writings from Ancient Greece mentioning pollution and issues with air quality) and current atmospheric measurements (used to continuously refine existing models), wouldn’t it be possible to establish, from an arbitrary baseline, the (relative) tolerable temperature increase and associated GHG concentration limits in the atmosphere? Surely, the oldest the baseline, the less data and the more reliance of past climate models (adding noise and uncertainty). What do I miss?

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(After receiving consent, I am moving here a comment made to me by @sarahshannon (thank you!), see quoted text below):

The choice of baselines from which to calculate climate change can sometimes have an impact on the interpretation of predictions. It is often pragmatic to use a 19th century baseline so that climate models can be evaluated against observations.

Typically, a period longer than 30 years is recommended to capture the natural inter-annual variability in the climate.

I came across this interesting paper.
The authors made river flow predictions for 8 basins for the end of the century. For some of their predictions they found opposite trends in river flow when using two different baselines (1961–1990) and (1986–2005). Both baselines are relatively recent, however, it’s interesting to see the diverging trends.

I can totally see situations in which the choice of baseline is very important. For every question of the form “How has X evolved?” the answer will almost certainly depend on the time period selected (the baseline being the lower end of the selected time interval). For instance, GHG emissions have dropped in the Covid period, but zooming out, they kept on growing to new highs. However, when it comes to define some tipping points such as the tolerable temperature increase and associated GHG concentration limits in the atmosphere, cannot these be established relative to the chosen baseline? In other words, if we want to limit an increase in temperature to 1.5C from pre-industrial era (i.e. 1850-1900), isn’t there a way to translate this to another baseline? If others think 1720-1800 is a better baseline, then this 1.5C limit becomes 1.5C \pm \varepsilon, where \varepsilon is the “baseline change adjustment”.

Does that make sense or do I miss something obvious? I guess what I am trying to say is that, surely for some questions (e.g. study of evolution of a parameter within a specific time frame) the baseline matters, but for others (e.g. setting limits and boundaries) do baselines truly have an impact? (I am assuming all other things equal, i.e. for a given/fixed model).