stargazeruses the same dependent variable because your dependent variables are the same, and not because you are using the same statistical model. You may be interested in using the argument column.labels:
data(mtcars)
m0 <- lm(mpg ~ hp, data=mtcars)
m1 <- lm(mpg ~ wt, data=mtcars)
m2 <- glm(cyl ~ disp, data=mtcars)
stargazer(m0, m1, column.labels=c('foo','bar'), type="text")
stargazer(m0, m2, column.labels=c('foo','bar'), type="text")
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