Well, I guess I'm not sure what the "correct" behavior actually is here. prediction::prediction() (which is working under the hood for margins::cplot()) is not doing any any substantive transformation here, it's just wrapping base predict() with type = "response" so any output that you get from prediction() (and thus from cplot()) should - and does - match what you would get from just working directly with predict():

pred <- data.frame(expand.grid(var1 = c(0, 1), var2 = seq(min(data$var2), max(data$var2), 0.5)))
pred <- cbind(pred, predict(m, newdata = pred, type = "response", se.fit = TRUE))
pred$UL <- pred$fit + (1.96*pred$se.fit)
pred$LL <- pred$fit - (1.96*pred$se.fit)
ggplot(pred, aes(var2, fit, fill = factor(var1))) + 
  geom_line(aes(col = factor(var1))) + 
  geom_ribbon(aes(x = var2, ymin = LL, ymax = UL), alpha = 0.4)

This is also equivalent to Stata's behavior on these example data:

use data.dta, clear
quietly logit outcome c.var1##c.var2
quietly margins, at(var2 = (-8(0.5)28) var1 = (0 1))
marginsplot

I suppose it's reasonable to expect a different output (akin to what you describe using plogis() and some googling suggests a fair amount of debate on what these intervals are and what you might want in a given circumstance. I'm totally open to suggestions on better behavior but as written, prediction() simply tidies the output of predict() rather than substantively modifying that output. If cplot() should do something different from the base behavior, I'm open to creating options for that, including changing the default.

I think the problem here is that type = "response" gives you the predicted probabilities, while type = "link" gives you the predictions on the log-odds scale (see ?predict.glm). Thus, the formula pred$fit +/- (1.96*pred$se.fit) won't work for predicted probabilities, as you need to back transform predictions and se, then do +/- 1.96*se and then back-back-transform. Or, you use type = "link", then you need to back-transform only once, after doing +/- 1.96*se. This ensures that predictions and CI are always within the correct (=possible) range.

See a similar discussion here: glmmTMB/glmmTMB#324 (comment)