Let me start by grounding this. In a scientific situation, people typically have theories about important relationships between variables. These theories are, however, incomplete in some way, or are not universally accepted. Thus, it is worthwhile to test specific hypotheses derived from those theories. The point of designing a study is to create a context in which there will be sufficient data that are relevant to that question. The point of building a regression model is to create a statistical context in which the hypothesis can be tested. This is the kind of situation that I typically work with, and what I generally have in mind when answering questions on CV.
At this point, let's say we've tested some treatment (intervention, exposure, etc.) effect and gotten a result. The model constitutes making a whole host of assumptions (not just independence, homoscedasticity, and normality). One might be concerned that the result we have is contingent on one or another assumption that makes our conclusion fragile. In that case, one way to proceed is to rerun the analysis / fit a new model that is robust to that kind of violation. It is common to call this a 'sensitivity analysis', although I don't know of this being an official definition anywhere. Here are some example write ups that might come out of this process:
... A concern is that our result could be driven by two high-leverage datapoints. As a sensitivity analysis, we refit our model without those points. The treatment remains significant and the mean difference is similar (-.51 vs. -.72).
... A concern is that our result could be driven by two high-leverage datapoints. As a sensitivity analysis, we fit a robust regression model using Tukey's bisquare as the loss function. The treatment remains significant and the mean difference is similar (-.51 vs. -.72).
... A concern is that the residuals may not be sufficiently normal to rely on normal theory to determine the p-values. As a sensitivity analysis, we bootstrapped the residuals to compute the p-value without assuming normality. The treatment remains significant (p=0.0031).
... We treated the summed score from the questionnaire as sufficiently equal-interval to use standard linear regression methods, because the scores are not against the bounds of the scale and this facilitates easier interpretation. However, one could argue that this is too cavalier for the statistical test of our primary hypothesis. As a sensitivity analysis, we replicated this result with an ordinal logistic regression (proportional odds) model. The intervention is significant in this model as well (p<0.02).
... Complete case analysis is valid under the assumption that the missingness is MAR. We have argued that this assumption is reasonable in our case. However, as a sensitivity analysis, we used the largest change observed in our dataset and assigned it, in the 'incorrect' direction, to those follow up visits that were missed. The question then is, how many do we have to add before the treatment effect becomes nonsignificant? ...
Etc. The above are the kind of thing that I think of when I see the phrase 'sensitivity analysis'. Note that @mdewey's meta-analysis example falls within this continuum.
On the other hand, no terms are all that well standardized. The same terms are used different ways by different fields. The Wikipedia definition sounds like error propagation in reverse. You start with a certain amount of uncertainty in the model's output, and "apportion" that to uncertainty in different inputs. (That is, '26% of the uncertainty is due to measurement error in X1', etc.) That is perfectly reasonable when developing a simulation model of a complex system, but it really doesn't fall within the category of analyses illustrated by the examples above.