Mohammed..
First of all, the 2nd projection is not written in one line. This can't work.
Secondly, and as said multiple times before, you need to validate your projection with gdalsrsinfo! Both of your projections are not valid (see gdalsrsinfo test below)!
I cannot follow why the 1st one works for you. They both don't work for me (see gdaltransform test below).
However, you still need to validate it to properly work with gdal.
gdalsrsinfo -V 'PROJCRS["ProjWiz_Custom_Cylindrical_Equal_Area",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563, LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Lambert Cylindrical Equal Area (Spherical)",ID["EPSG",9834]],PARAMETER["Latitude of 1st standard parallel",6.3630421,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Longitude of natural origin",53.9222717,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8802]],PARAMETER["False easting",0,LENGTHUNIT["metre",1],ID["EPSG",8806]],PARAMETER["False northing",0,LENGTHUNIT["metre",1],ID["EPSG",8807]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]'
ERROR 1: ERROR - failed to load SRS definition from PROJCRS["ProjWiz_Custom_Cylindrical_Equal_Area",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563, LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Lambert Cylindrical Equal Area (Spherical)",ID["EPSG",9834]],PARAMETER["Latitude of 1st standard parallel",6.3630421,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Longitude of natural origin",53.9222717,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8802]],PARAMETER["False easting",0,LENGTHUNIT["metre",1],ID["EPSG",8806]],PARAMETER["False northing",0,LENGTHUNIT["metre",1],ID["EPSG",8807]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]
gdalsrsinfo -V 'PROJCRS["ProjWiz_Custom_Albers",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563,LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Albers Equal Area",ID["EPSG",9822]],PARAMETER["Latitude of false origin",15.5248447,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8821]],PARAMETER["Longitude of false origin",48.1695557,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8822]],PARAMETER["Latitude of 1st standard parallel",13.2047435,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Latitude of 2nd standard parallel",17.844946,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8824]],PARAMETER["Easting at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8826]],PARAMETER["Northing at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8827]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]'
ERROR 1: ERROR - failed to load SRS definition from PROJCRS["ProjWiz_Custom_Albers",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563,LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Albers Equal Area",ID["EPSG",9822]],PARAMETER["Latitude of false origin",15.5248447,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8821]],PARAMETER["Longitude of false origin",48.1695557,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8822]],PARAMETER["Latitude of 1st standard parallel",13.2047435,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Latitude of 2nd standard parallel",17.844946,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8824]],PARAMETER["Easting at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8826]],PARAMETER["Northing at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8827]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]
gdaltransform -s_srs 'EPSG:4326' -t_srs 'PROJCRS["ProjWiz_Custom_Cylindrical_Equal_Area",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563, LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Lambert Cylindrical Equal Area (Spherical)",ID["EPSG",9834]],PARAMETER["Latitude of 1st standard parallel",6.3630421,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Longitude of natural origin",53.9222717,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8802]],PARAMETER["False easting",0,LENGTHUNIT["metre",1],ID["EPSG",8806]],PARAMETER["False northing",0,LENGTHUNIT["metre",1],ID["EPSG",8807]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]'
ERROR 1: Translating source or target SRS failed:
gdaltransform -s_srs 'EPSG:4326' -t_srs 'PROJCRS["ProjWiz_Custom_Albers",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563,LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Albers Equal Area",ID["EPSG",9822]],PARAMETER["Latitude of false origin",15.5248447,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8821]],PARAMETER["Longitude of false origin",48.1695557,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8822]],PARAMETER["Latitude of 1st standard parallel",13.2047435,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Latitude of 2nd standard parallel",17.844946,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8824]],PARAMETER["Easting at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8826]],PARAMETER["Northing at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8827]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]'
ERROR 1: Translating source or target SRS failed:
PROJCRS["ProjWiz_Custom_Albers",BASEGEOGCRS["WGS 84",DATUM["World Geodetic System 1984",ELLIPSOID["WGS 84",6378137,298.257223563,LENGTHUNIT["metre",1]],ID["EPSG",6326]],PRIMEM["Greenwich",0,ANGLEUNIT["Degree",0.0174532925199433]]],CONVERSION["unnamed",METHOD["Albers Equal Area",ID["EPSG",9822]],PARAMETER["Latitude of false origin",15.5248447,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8821]],PARAMETER["Longitude of false origin",48.1695557,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8822]],PARAMETER["Latitude of 1st standard parallel",13.2047435,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Latitude of 2nd standard parallel",17.844946,ANGLEUNIT["Degree",0.0174532925199433],ID["EPSG",8824]],PARAMETER["Easting at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8826]],PARAMETER["Northing at false origin",0,LENGTHUNIT["metre",1],ID["EPSG",8827]]],CS[Cartesian,2],AXIS["(E)",east,ORDER[1],LENGTHUNIT["metre",1,ID["EPSG",9001]]],AXIS["(N)",north,ORDER[2],LENGTHUNIT["metre",1,ID["EPSG",9001]]]]