Problem of projections and accuracy of measurements:
Since EPSG:3857 (and every Mercator projection) heavily distorts lengths (the more so the closer you get to the poles), measurements in this CRS don't make any sense, as @Vince already stated.
Cartesian vs. ellipsoidal measurements: which one is more accurate:
This also means that the difference between Cartesian and ellipsoidal method in your measurements are indeed due to the CRS you selected. The correct one (or better: closer to correct one) is indeed the ellipsoidal one as it does not measure distances projected to a flat canvas, but on the ellipsoid that the CRS is based on.
Problems of accuracy with ellipsoidal measurements:
But here exactly the problem starts. Not only the projection, but also the underlying ellipsoid can be responsible for a limited accuracy. It is, in fact, a model of Earth's shape, too and thus not 100% accurate. Even closer to Earth's shape is the Geoid - but again, this is a model too. On a next accuracy level, there is also topography to consider etc.
What you should consider for measuring:
You have to think about if the ellipsoid that EPSG:3857 is based on (WGS84) is accurate enough for what you intend to do. If you don't care about a few meters, it should be OK. Using a GPS receiver of a normal cell phone for example, you anyhow won't get enough accurate positioning (depending on model you use and the conditions/context where and when you measure up to +/- 15 meters). Accurate enough for a tourist to find London Tower Bridge, but not far from acceptable for an engineer building a road. So it always depends on use case.
In this context, see the Twitter post by @Nyall Dawson, a preview to the next release of QGIS 3.20, announced for release on June 18, 2021:
the war on EPSG:4326 has begun! Now #QGIS gives a friendly descriptive
warning when a CRS based on an ensemble datum (like WGS 84) is used.
As can be seen on the animated screencast, selecting EPSG:3857 will result in a warning:
Limited accuracy of at best 2 meters