I had to go earlier. I see Jakobian has extensively discussed this with you now and have not read your conservation, so apologies if I'm just regurgitating what has already been said, but here's how I would approach the problem:
If $R$ is a ring whose underlying additive group is finite cyclic of prime order $p$, pick an additive generator $x$. The elements of $R$ are then $kx$ for $k=0,\dotsc,p-1$. Thus, there is a constant $c\in\{0,\dotsc,p-1\}$ s.t. $x^2=cx$. This determines the multiplicative structure for $(kx)(lx)=klcx$. If $c=0$, the ring has trivial multiplication. Otherwise, $c$ is…

but in general, if $X$ is cut out by $f_i=\sum_{j_0,\dotsc,j_n,\sum j_l=d_i}a_{j_0\dotsc j_n}^iT_0^{j_0}\cdot\dotsc\cdot T_n^{j_n}$ for $i=1,\dotsc,m$, the conditions on the line through $[x_0\colon\dotsc\colon x_n]$ and $[y_0\colon\dotsc\colon y_n]$ I get are that
$\sum_{j_0,\dotsc,j_n,\sum j_l=d_i}\sum_{k_0,\dotsc,k_n,\sum k_l=k}a_{j_0\dotsc j_n}^i{j_0\choose k_0}\cdot\dotsc\cdot{j_n\choose k_n}x_0^{k_0}\cdot\dotsc\cdot x_n^{k_n}y_0^{j_0-k_0}\cdot\dotsc\cdot y_n^{j_n-k_n}$ for $i=1,\dotsc,m$ and $k=0,\dotsc,d_i$, which is not quite enlightening