Here’s what I do, in orchestral scores for various composers/publishers.
The + sign is sufficient for bouché, so there’s no need, except when following a manuscript, to use the text.
Cuivré describes a sound quality, often but not always hand-stopped. Always use text for this.
For trumpets (and trombones)
if the straight mute is the only mute required, con sord./senza sord. are sufficient indications, since that is the standard type.
If there is more than one type of mute, place the type in parentheses after: con sord. (cup) , or con sord. (straight) , etc. (There’s never a need to specify mute type after senza sord. )
For reminders – to confirm that the mute is still on after a long rest – use parens: (con sord.) if there’s only straight mute in use, but place the mute type in parens if there’s more than one: (straight mute), (cup mute), (harmon mute, stem extended) .
I use the vernacular language for these (which is English for almost all scores I work on). These are only included if the muted entry is far enough away that it’s not on the same system.
When the straight mute is the only one in use, write mute in after the last unmuted note; write mute out after the last muted note if the next entry is far enough away to warrant it.
If there is more than one mute in use, the warnings for the insertion specify the type: cup mute in , or harmon mute in (stem removed).
If the mute type changes from one to another, the warning is change to cup mute.
This covers about 95% of what I do. Hope it helps.