Your idea of paired-end reads is correct. These are libraries of ~300-500 bp inserts where both ends are sequenced. So paired-end reads sequence inwards (
Mate-pair reads on the other hand are from special libraries where the insert is large. Library fragments are circularized so the two ends then come into proximity. This makes a new sequencable construct. (LINK) In this case you get to sample the ends of that large insert.
Long read libraries are specially made from large MW DNA that is carefully handled not to cause breaks so you end up with "libraries" with fragments that are Kb's (or even Mb long). Depending on success a technology like ONT will be able to sequence through that entire stretch of DNA in 5'-->3' direction.