Context. The period-luminosity diagram (PLD) has proven to be a powerful tool for studying populations of pulsating red giants. Gaia Data Release 2 (DR2) provides a large data set including many long-period variables (LPVs) on which this tool can be applied. Aims. We investigate the location of LPVs from the Large and Small Magellanic Clouds in the PLD using various optical and infrared luminosity indicators from Gaia and 2MASS, respectively. We thereby distinguish between stars of different masses and surface chemistry. Methods. The data set taken from the Gaia DR2 catalogue of LPVs allows for a homogeneous study from low- to high-mass LPVs. These sources are divided into sub-populations of asymptotic giant branch (AGB) stars according to their mass and their O- or C-rich nature using the Gaia-2MASS diagram developed by our group. This diagram uses a Wesenheit index W_{BP, RP} based on Wesenheit functions in the Gaia and 2MASS photometric bands. Four different luminosity indicators are used to study the period-luminosity (P–L) relations. Results. We provide the first observational evidence of a P–L relation offset for both fundamental and 1O pulsators between low- and intermediate-mass O-rich stars, in agreement with published pulsation predictions. Among the luminosity indicators explored, sequence C′ is the narrowest in the P–W_{BP, RP} diagram, and is thus to be preferred over the other PLDs for the determination of distances using LPVs. The majority of massive AGB stars and red supergiants form a smooth extension of sequence C of low- and intermediate-mass AGB stars in the P–W_{BP, RP} diagram, suggesting that they pulsate in the fundamental mode. All results are similar in the two Magellanic Clouds.