The reason your final T is not orthogonal is that your original L is not orthogonal ($L^T=L \neq L^{-1}$). In fact Lorentz boosts really are not orthogonal, and the matrix you have written for L is correct. If L were orthogonal it would leave $t^2+x^2$ invariant. Instead L leaves $t^2-x^2$ invarient.

The request to prove orthogonality in your question doesn't go with the real matrix L that you wrote. Instead orthogonality goes with $L_{Historical}$ rotating by an imaginary angle.

$$ \begin{bmatrix} x'\\ ict'\\ \end{bmatrix} =L_{Historical}\begin{bmatrix} x\\ ict\\ \end{bmatrix} = \begin{bmatrix} cos(i\lambda) & -sin(i\lambda)\\ sin(i\lambda) & cos(i\lambda)\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} = \begin{bmatrix} cosh(\lambda) & -i\ sinh(\lambda)\\ i\ sinh(\lambda) & cosh(\lambda)\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} $$ $$ \begin{bmatrix} x'\\ ict'\\ \end{bmatrix} = \begin{bmatrix} \gamma & -i\beta\gamma\\ i\beta\gamma & \gamma\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} $$

The reason we have heard that boosts are somehow orthogonal rotations is that old time physicists made boosts look like familiar rotations by using imaginary angles and making t imaginary. Please see this Physics Stack answer for a little more explanation.

The reason your final T is not orthogonal is that your original L is not orthogonal ($L^T=L \neq L^{-1}$). In fact Lorentz boosts of $(ct,x)$ really are not orthogonal, and the matrix you have written for L is correct. If L were orthogonal it would leave $t^2+x^2$ invariant. Instead L leaves $t^2-x^2$ invariant.

The request to prove orthogonality in your question doesn't go with the real matrix L that you wrote. Instead orthogonality goes with $L_{Historical}$ rotating by an imaginary angle.

$$ \begin{bmatrix} x'\\ ict'\\ \end{bmatrix} =L_{Historical}\begin{bmatrix} x\\ ict\\ \end{bmatrix} = \begin{bmatrix} cos(i\lambda) & -sin(i\lambda)\\ sin(i\lambda) & cos(i\lambda)\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} = \begin{bmatrix} cosh(\lambda) & -i\ sinh(\lambda)\\ i\ sinh(\lambda) & cosh(\lambda)\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} $$ $$ \begin{bmatrix} x'\\ ict'\\ \end{bmatrix} = \begin{bmatrix} \gamma & -i\beta\gamma\\ i\beta\gamma & \gamma\\ \end{bmatrix} \begin{bmatrix} x\\ ict\\ \end{bmatrix} $$

The reason we have heard that boosts are somehow orthogonal rotations is that old time physicists made boosts look like familiar rotations by using imaginary angles and making t imaginary. Please see this Physics Stack answer for a little more explanation.