Context. Mapping distant worlds is the next frontier for exoplanet infrared (IR) photometry studies. Ultimately, constraining spatial and temporal properties of an exoplanet atmosphere (e.g., its temperature) will provide further insight into its physics. For tidallylocked hot Jupiters that transit and are eclipsed by their host star, the first steps are now possible. Aims. Our aim is to constrain an exoplanet's (1) shape, (2) brightness distribution (BD) and (3) system parameters from its phase curve and eclipse measurements. In particular, we rely on the secondary-eclipse scanning which is obtained while an exoplanet is gradually masked by its host star. Methods. We use archived Spitzer/IRAC 8-µm data of HD 189733 (six transits, eight secondary eclipses, and a phase curve) in a global Markov chain Monte Carlo (MCMC) procedure for mitigating systematics. We also include HD 189733's out-of-transit radial velocity (RV) measurements to assess their incidence on the inferences obtained solely from the photometry. Results. We find a 6σ deviation from the expected occultation of a uniformly-bright disk. This deviation emerges mainly from a large-scale hot spot in HD 189733b's atmosphere, not from HD 189733b's shape. We indicate that the correlation of the exoplanet orbital eccentricity, e, and BD ("uniform time offset") does also depend on the stellar density, ρ , and the exoplanet impact parameter, b ("e-b-ρ -BD correlation"). For HD 189733b, we find that relaxing the eccentricity constraint and using more complex BDs lead to lower stellar/planetary densities and a more localized and latitudinally-shifted hot spot. We, therefore, show that the light curve of an exoplanet does not constrain uniquely its brightness peak localization. Finally, we obtain an improved constraint on the upper limit of HD 189733b's orbital eccentricity, e ≤ 0.011 (95% confidence), when including HD 189733's RV measurements. Conclusions. Reanalysis of archived HD 189733's data constrains HD 189733b's shape and BD at 8 µm. Our study provides new insights into the analysis of exoplanet light curves and a proper framework for future eclipse-scanning observations. In particular, observations of the same exoplanet at different wavelengths could improve the constraints on HD 189733's system parameters while ultimately yielding a large-scale time-dependent 3D map of HD 189733b's atmosphere. Finally, we discuss the perspective of extending our method to observations in the visible (e.g., Kepler data), in particular to better understand exoplanet albedos.