If you look at whether your confidence intervals are below $2$ or above $2$, of those from the exponential distribution about 11.3% are too low, 88.3% include the population mean of $2$, and 0.4% are too high, while with a normal distribution you would get about 2.5%, 95% and 2.5% respectively. You could use the following R code to simulate this.
The principal cause of this is that the exponential distribution is too skewed with a sample size of only $5$. The sample mean from an exponential distribution has a Gamma distribution which, (largely but not exclusively) because of its skewness, is poorly modelled by a Student's $t$ distribution.
cases <- 5000
ss <- 5
popmean <- 2
#sampledata <- matrix(rnorm(cases*ss, mean=popmean, sd=1), ncol=ss) #normal
#sampledata <- matrix(runif(cases*ss, min=0, max=2*popmean), ncol=ss) #uniform
sampledata <- matrix(rexp(cases*ss, rate=1/popmean), ncol=ss) #exp
samplemean <- rowMeans(sampledata)
samplese <- sqrt((rowSums(sampledata^2)/ss - samplemean^2)/(ss-1))
confintfactor <- qt(0.975, df=ss-1) # about 2.776 for ss=5
toolow <- sum( samplemean + confintfactor*samplese < popmean) / cases
toohigh <- sum( samplemean - confintfactor*samplese > popmean) / cases
c(toolow, 1-toolow-toohigh, toohigh) # would like 0.025 0.95 0.025
For the uniform distribution, about 3.3% of the confidence intervals are too low, 93.4% include the population mean of $2$, and 3.3% are too high. Here too the Student's $t$ distribution is not a perfect model, but it is better than before: in particular there is no skewness.