I don't know what software you are using to perform your analysis. I have calculated pooled estimate effect of mine by using R software. There I didn't calculate the logit odds. Simply I just put the number that would estimate the odds ratio and run the required test.
So, I guess there won't be any difference between these two.
Actually, most meta-analysis tools automatically convert to log odds ratios for the analysis and back for reporting. You don't have to calculate this yourself, but the tool most likely will. The log odds ratio is symmetric around zero and more closely approximates a normal distribution. This goes for metafor in R and Meta-Essentials in Excel - and I believe for RevMan and CMA also.
It depends on what software you are using. If it is R, you don't have to, but you have to indicate in the code the nature of the data you are using. In RevMan, you can use the calculator function to introduce the estimates of the effect size (ORs), and it'll be directly converted into logOR.