Using logistic regression for a continuous dependent variable I got a revision for my research paper recently and the following is the reviewer's comment on my paper:

results obtained from one model is not quite convincing especially linear regression usually has deficiencies in dealing with outliers. I suggest the authors also try logistic regression and compare the corresponding results with current results. If the similar observations are obtained, the results would be more solid.

Is the reviewer's comment right? Is logistic regression better than multiple linear regression?
The problem is that my dependent variable is not categorical, it's a scale variable. What can I do now? What other regression method do you recommend to evaluate my model?
Score is dependent variable in the following table. Recency, frequency, tenure and last score are independent variables.

I've extracted these variables from a site and I hypothesize that these independent variables have significant effect on the score. Therefore, I represent the following models:


By the way, the value of R squared for this linear model is 0.316! The reviewer also commented on this value too:

then the results are not convincing as there is no indicator on the quality of learned coefficients. A small R^2 cannot indicate good performance as the model may be over-fitted.

Is 0.316 very low for R squared? In previous papers I saw the similar values a lot.

 A: The proportional odds ordinal logistic regression model should work fine for this problem.  For an efficient implementation that can allow thousands of unique $Y$ values see the orm function in the R rms package.
A: you could also try ordered probit/logit models by assigning values 1, 2,3, and 4 to scores in the 1st,.....,4th percentiles respectively.
A: You could dichotomise (convert to a binary variable) the score.
If score is from 0 to 100 then you could assign 0 to any score less than 50 and 1 otherwise.
I've never heard before that this a good way of dealing with outliers though.  This might just hide outliers since it will be impossible to distinguish very high or low scores.  This doesn't make a great deal of sense to me but you can try it.
More importantly why are you log transforming all your covariates and your response variable?
This is going to affect your $\beta$ estimates and your $R^2$ (i think).
Also the reviewer says a small $R^2$ suggests overfitting?  I thought overfitting was when your $R^2$ is high but your model performs poorly on new data (i.e it overfits your data but doesn't generalise to new data).  Overfitting tends to happen when you have few observations which you are trying to predict with a large number of parameters.  This is what you are doing in your Model 2 since you have 8 observations which you are trying to explain with 7 parameters.
I am not going to pretend I know a great deal about statistics but it seems to me, based on his comments, that this reviewer might know even less.
A: It is possible to apply logistic regression even to a contiuous dependent variable. It makes sense, if you want to make sure that the predicted score is always within [0, 100] (I judge from your screenshots that it is on 100-point scale). 
To accomplish it, just divide your score by 100, and run logistic regression with this [0,1]- based target variable, like in this question - you can do it, for example, with R, using 
glm(y~x, family="binomial", data=your.dataframe)

I don't know whether this approach helps with outliers - it depends on the sort of outliers you are expecting. But sometimes it improves goodness of fit (even $R^2$, if your dependent variable has natural lower and upper bounds. 
As for the second question, $R^2\approx 0.3$ may be the best what you can squeeze out of your data, without overfitting. If you build your model for the purpose of inference, low $R^2$ is totally fine, as long as the coefficients important to you are significant. If you want to check whether the model is overfitted, you can check its $R^2$ on a test set, or even do a cross-validation.
