As I were implementing the pinhole camera model based on Tsai, I noticed that there were two conventions used. For opencv and others, the following is used:
$$ \begin{pmatrix} u_{c} \\\ v_{c} \\\ h_{c} \end{pmatrix} = R * P + T $$
$$ \begin{matrix} u_{n} = u_{c} / h_{c} & v_{n} = v_{c} / h_{c} \end{matrix} $$
$$ \begin{pmatrix} u \\\ v \\\ 1 \end{pmatrix} = K * \begin{pmatrix} u_{n} \\\ v_{n} \\\ 1 \end{pmatrix} $$
However, some researchers used the following definition:
$$ \begin{pmatrix} u_{c} \\\ v_{c} \\\ h_{c} \end{pmatrix} = R^{-1} * (P - T) $$
$$ \begin{matrix} u_{n} = u_{c} / h_{c} & v_{n} = v_{c} / h_{c} \end{matrix} $$
$$ \begin{pmatrix} u \\\ v \\\ 1 \end{pmatrix} = K * \begin{pmatrix} u_{n} \\\ v_{n} \\\ 1 \end{pmatrix} $$
Where
$$ P = \begin{pmatrix} X \\\ Y \\\ Z \end{pmatrix} $$
$$ R = \begin{pmatrix} r_1 & r_2 & r_3 \\ r_4 & r_5 & r_6 \\ r_7 & r_8 & r_9 \end{pmatrix} $$
$$ T = \begin{pmatrix} t_{x} \\\ t_{y} \\\ t_{z} \end{pmatrix} $$
$$ K = \begin{pmatrix} f_{x} & 0 & c_{x} \\\ 0 & f_{y} & c_{y} & \\\ 0 & 0 & 1 \end{pmatrix} $$
My problem is I do not know why the second definition is used. When using direct linear transform to estimate intrinsic and extrinsic parameters, the first definition is the only one I know that works. Furthermore, the first definition provides the correct $u$ and $v$ values for a given focal length, Euler angles, and translation vector. However, the second definition requires different values for the parameters and is harder to optimize through nonlinear minimization with error terms. The only reason why I can think of its use is the ease of which it is to back project points to a specified $Z$ plane. So, does anyone know why does this definition exists and can provide more information on its use?
P.S. Sorry for any ambiguity in my question as this is the first time using math.exchange and second time using forums for questioning at all. I searched for many definitions in the forums and other websites, but could not get any conclusive reasons why there are different definitions for perspective projection using the pinhole camera model.