As you have recognized, the difference in the coefficients is related to the link functions. The inverse link functions results in an inverse function of the linear predictor and therefore leads to a negative coefficient when the relationship between dependent and independent variable is positive.
To decide which model family to use, I would suggest using the DHARMa package to see which model fits best. It has a great vignette which will guide you through residual diagnostics for generalized linear mixed models.
You can also compare the residual deviance of your models with anova(m1, m2) and their parsimony with AIC(m1, m2) (although anova will also include AIC and BIC, which are indicators of residual deviance weighted by model complexity, lower values indicating lower residual deviance and/or simpler models, which are preferable). However, this part should not supplant residual diagnostics which are essential to see if the models fit the data.