Centroid coordinates for odd-shaped polygons
I think the standard QGIS polygon centroid tools calculate the centre of mass in order to determine the polygon's centroid. So this could lie outside the polygon itself (nothing to do with projection).
If you want the calculate the centroid in terms of its dimensions, you could use the realcentroid plugin instead which you can download from:
Plugins > Manage and Install Plugins...
Example:
An irregular polygon with the red circle calculated from the Polygon centroid tool from QGIS (and SAGA); the green circle calculated from the realcentroid plugin:
There is also an online script which produces the same result as the plugin which you can download from:
Processing Toolbox > Scripts > Tools > Get scripts from on-line scripts collection
And find the Real centroid script.
Proessing QGIS Algorithms has "Point on Surface" too, under Vector geometry tools group. No need to use external plugins nor DBs