How to describe the method to compute standard errors of a difference?

Question

I wonder how to name or describe some methods of computing the standard error $se$ or confidence interval $ci$ of a difference (or sum or whatever). For example, if the bootstrap method is used I could write: The 95%-CIs for the difference in 5-year risk were obtained by the percentile method from 1000 bootstrap iterations.

The description may be written in a paper and addressed to a medical audience.

Now, there are two other methods to compute standard errors:

1. Standard deviations are given with degrees of freedom: $\sigma_1, df_1,\sigma_2,df_2$
   First compute the pooled variance: $\sigma_p=\frac{df_1\sigma_1^2+df_2\sigma_2^2}{df_1+df_2}$
   Then compute the standard error: $se=\sqrt{\sigma_p(\frac{1}{n_1}+\frac{1}{n_2})}$

2. Only standard errors are given: $se_1,se_2$
   $se=\sqrt{se_1^2+se_2^2}$

I'm wondering how to name or describe these two methods. I could describe them

1. Standard error is computed using the pooled variances of the two samples considering degree of freedoms.
2. Standard error is computed pooling the standard errors.

Are there any established descriptions for these methods? Any suggestions?