The odds of having a diagnosis in model 2 decreases by a factor 0.91 for a unit increase in score B. Some people don't like these numbers less than 1, so they take the invers. This is now interpreted that the odds of a diagnosis increases by a factor 1,10 for a unit decrease in factor B.
If you multiply something by 1.10, you increase it by 10%. In general the relationship between a factor increase and the percentage change is (f - 1) * 100%. So an odds ratio of 0.91 corresponds to a (0.91 - 1)*100% = -9% change in odds for a unit increase in factor B, or a 9% decrease in the odds for a unit change in factor B.
Ordinarily, we interpret coefficients in terms of how the expected value of the response should change when we effect tiny changes in the underlying variables. This is done by differentiating the formula, which is
$$E\left[\log Y\right] = \beta_0 + \beta_1 x_1 + \beta_2\left(\frac{x_3}{x_1}\right).$$
The derivatives are
$$\frac{\partial}{\partial x_1} E\left[\log Y \right] = \beta_1 - \beta_2\left( \frac{x_3}{x_1^2}\right)$$
and
$$\frac{\partial}{\partial x_3} E\left[\log Y \right] = \beta_2 \left(\frac{1}{x_1}\right).$$
Because the results depend on the values of the variables, there is no universal interpretation of the coefficients: their effects depend on the values of the variables.
Often we will examine these rates of change when the variables are set to average values (and, when the model is estimated from data, we use the parameter estimates as surrogates for the parameters themselves). For instance, suppose the mean value of $x_1$ in the dataset is $2$ and the mean value of $x_3$ is $4.$ Then a small change of size $\mathrm{d}x_1$ in $x_1$ is associated with a change of size
$$\left(\frac{\partial}{\partial x_1} E\left[\log Y \right] \right)\mathrm{d}x_1 = (\beta_1 - \beta_2(4/2^2))\mathrm{d}x_1 = (\beta_1 - \beta_2)\mathrm{d}x_1.$$
Similarly, changing $x_3$ to $x_3+\mathrm{d}x_3$ is associated with change of size
$$\left(\frac{\partial}{\partial x_3} E\left[\log Y \right] \right)\mathrm{d}x_3 = \left(\frac{\beta_{2}}{2}\right)\mathrm{d}x_3$$
in $E\left[\log y\right].$
For more examples of these kinds of calculations and interpretations, and to see how the calculations can (often) be performed without knowing any Calculus, visit How to interpret coefficients of angular terms in a regression model?, How do I interpret the coefficients of a log-linear regression with quadratic terms?, Linear and quadratic term interpretation in regression analysis, and How to interpret log-log regression coefficients for other than 1 or 10 percent change?.
Best Answer
Let's look at what's going on behind the scenes. A logistic regression model (binomial GLM) looks like this: $$ \log\left(\frac{\pi}{1-\pi}\right)=\beta_0+\beta_1X_1 $$ assuming we're only fitting an intercept and one parameter for your $X_1$ variable.
As you correctly state, exponentiating the coefficients gives us the odds: $$ \exp\left[\log\left(\frac{\pi}{1-\pi}\right)\right]=\exp(\beta_0+\beta_1X_1)\\ \implies \frac{\pi}{1-\pi} = \exp(\beta_0)\exp(\beta_1X_1) $$ For simplicity, let's focus on $\beta_1$ and assume that $\beta_0=0$. This makes $\exp(\beta_0)=\exp(0)=1$, and therefore the above expression reduces to: $$ \frac{\pi}{1-\pi} = \exp(\beta_1X_1) $$ Now, let's look at what happens when $X_1$ increases by one unit e.g. from $2$ to $3$. Let's also imagine $\beta_1=-0.74$: $$ (1) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times2)=\exp(-1.48)\approx0.23 \\ (2) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times3)=\exp(-2.22)\approx0.11 $$ A unit increase in $X_1$ has resulted in a decrease of $0.11/0.23=0.478$ in the odds. Note that this is just the odds ratio, and $\exp(\beta_1)=\exp(-0.74)=0.478$. So a unit increase in $X_1$ results in the odds of success being multiplied by the odds ratio, which is nothing but $\exp(\beta_1)$. Essentially, if $\beta_1>0$, an increase in $X_1$ will yield higher odds (because the odds ratio > 1), whereas if $\beta_1<0$, an increase in $X_1$ will yield lower odds (because the odds ratio < 1).
Now, you should be able to see for yourself what would happen if we were working with $\ln(X_1)$ rather than with $X_1$. Hint: $$ (3) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times\ln(2))=2^{-0.74}\approx0.60\\ (4) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times\ln(3))=3^{-0.74}\approx0.44 $$ That is, a decrease in the odds of $0.44/0.60=0.74$. However, the odds ratio is still $\exp(\beta_1)=\exp(-0.74)=0.478$, so what's going on here?
By taking the natural logarithm of $X_1$, you change the interpretation of its coefficient. Now, the odds ratio represents the change in the odds if you multiply $X_1$ by $e$. Therefore: $$ (5) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times\ln(2))=2^{-0.74}\approx0.599\\ (6) \space\space\frac{\pi}{1-\pi} = \exp(-0.74\times\ln(2\times e))=(2\times e)^{-0.74}\approx0.286 $$ We see that multiplying $X_1$ by $e$ results in the odds decreasing from $0.599$ to $0.286$. Here $0.286/0.599=0.478$ which is precisely the odds ratio i.e. $\exp(\beta_1)=\exp(-0.74)$.
Following the same logic, it's easy to see that: