Any formula in physics comes with a set of definitions of what each variable in the equation represents, and how to interpret positive or negative values. This is particularly rue in the case of lens and mirror formulae. In each case, a different form of the equation, with a different set of definitions, will give the same correct result.
In this case, Wikipedia https://en.wikipedia.org/wiki/Lens_(optics)#Lensmaker.27s_equation adds to the above equation:
The signs of the lens' radii of curvature indicate whether the
corresponding surfaces are convex or concave. The sign convention used
to represent this varies, but in this article a positive R indicates a
surface's center of curvature is further along in the direction of the
ray travel (right, in the accompanying diagrams), while negative R
means that rays reaching the surface have already passed the center of
curvature. Consequently, for external lens surfaces as diagrammed
above, R1 > 0 and R2 < 0 indicate convex surfaces (used to converge
light in a positive lens), while R1 < 0 and R2 > 0 indicate concave
surfaces. The reciprocal of the radius of curvature is called the
curvature. A flat surface has zero curvature, and its radius of
curvature is infinity
Thus, your "on the other hand" individual is not respecting the sign convention, and will not get the correct result...
EDIT to expand
This source, http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html, with its links, define exactly how the direction of light flow, sign of radius of curvature and position of focal point are to be defined. If you fail to follow these conventions with this form of the equation, chaos results...