= Solution
For parameter of interest $\psi$ and nuisance parameter $\lambda$, the <profile likelihood> has log form $\ell_p(\psi)=\ell(\psi,\widehat\lambda_\psi)$, where $\widehat\lambda_\psi$ is the constrained nuisance maximum. Profiling optimizes over the nuisance dimension but does not account for its local volume or conditioning geometry. The <modified profile likelihood> corrects this loss.
Use ancillary sample coordinates $(\widehat\psi,\widehat\lambda,a)$, and let $j_{\lambda\lambda}(\psi,\widehat\lambda_\psi)=-\ell_{\lambda\lambda}$ be the nuisance block of <observed information>. Holding $\widehat\psi,a$ fixed, put $D(\psi)=|\partial\widehat\lambda_\psi/\partial\widehat\lambda|$. The <Barndorff-Nielsen modified profile likelihood> is, up to data-only factors,
$$
\boxed{\ell_m(\psi)=\ell_p(\psi)-\frac12\log|j_{\lambda\lambda}(\psi,\widehat\lambda_\psi)|-\log D(\psi).}
$$
This combines a nuisance-information determinant and a data-coordinate <Jacobian determinant>. Implicit differentiation of $\ell_\lambda(\psi,\widehat\lambda_\psi)=0$ gives $\partial\widehat\lambda_\psi/\partial\widehat\lambda=j_{\lambda\lambda}^{-1}\ell_{\lambda;\widehat\lambda}$, so equivalently
$$
\ell_m(\psi)=\ell_p(\psi)+\frac12\log|j_{\lambda\lambda}|-\log|\ell_{\lambda;\widehat\lambda}|.
$$
The semicolon denotes differentiation in the varying sample coordinate, rather than a parameter derivative. This form also explains the positive adjustment in Question 2: in that canonical conditional problem the mixed sample derivative is data-only, leaving the positive half-log curvature. The adjustment sign cannot be copied between unrelated constructions without tracking their Jacobians.
In an appropriate orthogonal nuisance formulation, the <Cox-Reid adjusted profile likelihood> uses $\ell_p-\tfrac12\log|j_{\lambda\lambda}|$. It is a useful simpler adjustment, but discarding $D$ is not universally legitimate. For normal data with interest $\sigma^2$ and nuisance $\mu$, the constrained and full nuisance fits both equal $\bar X$, so $D=1$ and $j_{\mu\mu}=n/\sigma^2$. Thus
$$
\ell_m(\sigma^2)=-\frac{n-1}{2}\log\sigma^2-\frac{\sum_i(X_i-\bar X)^2}{2\sigma^2}+\text{constant},
$$
recovering the residual degrees of freedom and fitted variance $\sum_i(X_i-\bar X)^2/(n-1)$. These adjustments target nuisance-related bias and higher-order likelihood inference; they are approximations to conditional or marginal constructions, not an assertion that profiling has become an exact full sampling density.
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