Actually, when I google martyr, the definition is simply someone who is killed for their religious beliefs.Really? What hope is being kindled here? What honour? Being dragon chow? Senselessly throwing away your life to accomplish nothing? For a paladin whose oath directly states that you should revere life above all things? You talk about faith, but, ignore the actual faith of the paladin in question.
I get the point about martyrdom, I do. But, thing is, there needs to be some sort of a cause to be a martyr to. Some symbol besides, "You are crunchy and taste good with ketchup". This is a pointless death, not a martyrdom.
Becoming a martyr is more than just committing suicide by dragon. Otherwise your world must have an awful lot of martyrs.
My bet is there are as many implied meanings for martyr as you have Faith's and beliefs and causes.
Part of the frustration of some of these positions which shift away from the context of 5e oaths and alignment to real world philosophical treatises and faiths is that it seems like from one moment to the next the target shifts to whatever philosopher refutes any claim.
To me, honestly, much of these if actually thrown in play redult in a character functionally unplayable as a PC in a troupe game that's at all like 5e presents. They may be more well suited for a one-on-one game where the setting and systems are built around these concepts.
But, if the GM puts forth that paladins are honor bound to lay down their lives against new threats as soon as they start to protect or help escape any NPC - my bet is that is effectively reducing paladins to NPCs because I doubt I have seen a single party ready to sign that guy up.