An E-1 connection just indicates its bandwidth capacity. Max speed of E-1 is usually 2 Mbps.
Normally, when a line is "leased", it usually refers to a dedicated line end-to-end (i.e. attached to each of your sites), that's always connected. This could be an E-1, or a line that's smaller or larger in its bandwidth capacity.
A "leased line" is in contrast to "normal" usage of the PSTN (public switched telephone network) where you obtain a connection for the duration of your call.
You can also obtain leased lines to various cloud technologies, such as frame-relay, ATM and MPLS. These are usually referred to by their technology. E.g. an ATM link. Further these lines could also be E-1 capacity, or smaller or larger too.
