Here is a variation of Lucia's nice argument. Writing $$|d(n)-\log n|=d(n)+\log n-2\min(d(n),\log n),$$ we see that $$R(x)=2x\log x+O(x)-2\sum_{n\leq x}\min(d(n),\log n).$$ Now let $\kappa\in(0,1)$ be fixed, and let us use that $\min(d(n),\log n)\leq d(n)^\kappa(\log n)^{1-\kappa}$. Then, by Theorem 2 in Chapter II.6 of Tenenbaum: Introduction to analytic and probabilistic number theory, we get $$\sum_{n\leq x}\min(d(n),\log n)\leq(\log x)^{1-\kappa}\sum_{n\leq x}2^{\kappa\Omega(n)}\ll_\kappa x(\log x)^{2^\kappa-\kappa}.$$ The value $\kappa:=-\log\log 2/\log 2$ yields the minimal exponent $$2^\kappa-\kappa=(1+\log\log 2)/\log 2\approx 0.914,$$ whence $$R(x)=2x\log x + O\left(x(\log x)^{(1+\log\log 2)/\log 2}\right).$$

Here is a variation of Lucia's nice argument. Writing $$|d(n)-\log n|=d(n)+\log n-2\min(d(n),\log n),$$ we see that $$R(x)=2x\log x+O(x)-2\sum_{n\leq x}\min(d(n),\log n).$$ Now let $\kappa\in(0,1)$ be fixed, and let us use that $\min(d(n),\log n)\leq d(n)^\kappa(\log n)^{1-\kappa}$. Then, by Theorem 2 in Chapter II.6 of Tenenbaum: Introduction to analytic and probabilistic number theory, we get $$\sum_{n\leq x}\min(d(n),\log n)\leq(\log x)^{1-\kappa}\sum_{n\leq x}2^{\kappa\Omega(n)}\ll_\kappa x(\log x)^{2^\kappa-\kappa}.$$ The value $\kappa:=-\log\log 2/\log 2\approx 0.529$ yields the minimal exponent $$2^\kappa-\kappa=(1+\log\log 2)/\log 2\approx 0.914,$$ whence $$R(x)=2x\log x + O\left(x(\log x)^{(1+\log\log 2)/\log 2}\right).$$