How do I sort the texture positions based on the texture indices given in a Wavefront (.obj) file?

If there are different indexes for vertex coordinates and texture coordinates, then the vertex positions must be “duplicated”.
The vertex coordinate and its attributes (like texture coordinate) form a tuple. Each vertex coordinate must have its own texture coordinates and attributes. You can think of a 3D vertex coordinate and a 2D texture coordinate as a single 5D coordinate.
See Rendering meshes with multiple indices.

Let’s assume that you have a .obj file like this:

v -1 -1 -1
v  1 -1 -1
v -1  1 -1
v  1  1 -1
v -1 -1  1
v  1 -1  1
v -1  1  1
v  1  1  1 

vt 0 0
vt 0 1
vt 1 0
vt 1 1

vn -1  0  0 
vn  0 -1  0
vn  0  0 -1
vn  1  0  0
vn  0  1  0
vn  0  0  1

f 3/1/1 1/2/1 5/4/1 7/3/1
f 1/1/2 2/2/2 3/4/2 6/3/2
f 3/1/3 4/2/3 2/4/3 1/3/3
f 2/1/4 4/2/4 8/4/4 6/3/4
f 4/1/5 3/2/5 7/4/5 8/3/5
f 5/1/6 6/2/6 8/4/6 7/3/6

From this you have to find all the combinations of vertex coordinate, texture texture coordinate and normal vector indices, which are used in the face specification:

 0 : 3/1/1 
 1 : 1/2/1
 2 : 5/4/1
 3 : 7/3/1
 4 : 1/1/2
 5 : 2/2/2
 6 : 3/4/2
 7 : 6/3/2
 8 : ...

Then you have to create a vertex coordinate, texture coordinate and normal vector array corresponding to the array of index combinations.
The vertex coordinates and its attributes can either be combined in one array to data sets, or to three arrays with equal number of attributes:

 index   vx vy vz     u v     nx ny nz
 0 :     -1  1 -1     0 0     -1  0  0
 1 :     -1 -1 -1     0 1     -1  0  0
 2 :     -1 -1  1     1 1     -1  0  0
 3 :     -1  1  1     1 0     -1  0  0
 4 :     -1 -1 -1     0 0      0 -1  0
 5 :      1 -1 -1     0 1      0 -1  0
 6 :     -1  1 -1     1 1      0 -1  0
 7 :      1 -1  1     1 0      0 -1  0
 8 : ...

---

See the very simple c++ function, which can read an .obj file, like that you linked to.
The function reads a file and writes the data to an element vector and an attribute vector.

Note, the function can be optimized and does not care about performance.
For a small file (like cube3.obj which you liked to), that doesn’t matter, but for a large file,
especially the linear search in the index table, will have to be improved.

I just tried to give you an idea how to read an .obj file and how to create an element and attribute vector, which can be directly used to draw an mesh with the use of OpenGL.

#include <vector>
#include <array>
#include <string>
#include <fstream>
#include <strstream>
#include <algorithm>

bool load_obj( 
    const std::string          filename, 
    std::vector<unsigned int> &elements,
    std::vector<float>        &attributes )
{
    std::ifstream obj_stream( filename, std::ios::in );
    if( !obj_stream )
        return false;

    // parse the file, line by line
    static const std::string white_space = " \t\n\r";
    std::string token, indices, index;
    float value;
    std::vector<float> v, vt, vn;
    std::vector<std::array<unsigned int, 3>> f;
    for( std::string line; std::getline( obj_stream, line ); )
    {
        // find first non whispce characterr in line
        size_t start = line.find_first_not_of( white_space );
        if ( start == std::string::npos )
            continue;

        // read the first token
        std::istringstream line_stream( line.substr(start) );
        line_stream.exceptions( 0 );
        line_stream >> token;

        // ignore comment lines
        if ( token[0] == '#' )
            continue;

        // read the line
        if ( token == "v" ) // read vertex coordinate
        {
            while ( line_stream >> value )  
                v.push_back( value );
        }
        else if ( token == "vt" ) // read normal_vectors 
        {
            while ( line_stream >> value )
                vt.push_back( value );
        }
        else if ( token == "vn" )  // read normal_vectors 
        {
            while ( line_stream >> value )
                vn.push_back( value );
        }
        else if ( token == "f" )
        {
            // read faces
            while( line_stream >> indices )
            {
                std::array<unsigned int, 3> f3{ 0, 0, 0 };
                // parse indices
                for ( int j=0; j<3; ++ j )
                {
                    auto slash = indices.find( "/" );
                    f3[j] = std::stoi(indices.substr(0, slash), nullptr, 10);
                    if ( slash == std::string::npos )
                        break;
                    indices.erase(0, slash + 1);
                }
            
                // add index
                auto it = std::find( f.begin(), f.end(), f3 );
                elements.push_back( (unsigned int)(it - f.begin()) );
                if ( it == f.end() )
                    f.push_back( f3 );
            }
        }
    }

    // create array of attributes from the face indices
    for ( auto f3 : f )
    {
        if ( f3[0] > 0 )
        {
            auto iv = (f3[0] - 1) * 3;
            attributes.insert( attributes.end(), v.begin() + iv, v.begin() + iv + 3 );
        }

        if ( f3[1] > 0 )
        {
            auto ivt = (f3[1] - 1) * 2;
            attributes.insert( attributes.end(), vt.begin() + ivt, vt.begin() + ivt + 2 );
        }

        if ( f3[2] > 0 )
        {
            auto ivn = (f3[2] - 1) * 3;
            attributes.insert( attributes.end(), vn.begin() + ivn, vn.begin() + ivn + 3 );
        }
    }

    return true;
}