Sometimes in playing the guitar I do not unfret a note followed by a rest (typically a rest between two verses of a song) until the next one, letting it decay slowly. What is the official name for such an augmented duration of the note (not really just augmented because it is decaying). How to note this in a score using traditional notation? Is this something specific to the guitar or some of you do the same in other instruments as well? Is this the same as a sustained note, marked with a . and a bow above it? I do not think so, because I think the sustained note should not decay slowly and you need the piano sustain pedal for that. On the guitar you cannot even make it without some sort of special equipment ASAIK. So I think sustained and held notes are a quite different thing, am I right?
Either a hanging tie (that is, unattached to a following note) or l.v. (laissez vibrer), or both. This is used for other non-sustaining instruments (piano, harp, vibraphone, etc.) as well.
The fermata (the "bird's eye") means holding the duration longer than notated (an unmeasured interval of time), not just allowing it to decay - it applies to sustaining instruments as well as non-sustaining.
In guitar music specifically, the text annotation "let ring" along with an optional dashed line to indicate the span of music where this is desired is the typical notation.