Tarnish started out as something metal does on its own. Etymonline dates it to the mid-1400s in the sense "lose luster, become discolored by exposure to air," from Old French ternir, "to make dim," and Wiktionary still lists that intransitive sense first: store silver badly and it tarnishes with nobody's help. The transitive sense, one thing tarnishing another, only turns up in the 1670s, and the figurative "take away the purity of" in the 1690s.
Besmirch never had a version like that. It arrives in the 1590s as be- plus smirch, "to soil with soot or mud," and both senses Wiktionary gives are transitive. The line Etymonline quotes is from Henry V, just before Agincourt, where Henry tells the French herald his army's finery is a mess from marching in the rain: "Our gayness and our gilt are all besmirch'd / With rainy marching in the painful field." That's literal mud on literal gilt. Where smirch itself comes from is less clear. The OED's guess, which Etymonline passes along as uncertain, is Old French esmorcher, "to torture," built on a verb for biting from the same Latin mordere that gave us mordant.
So when it's a reputation, I pick by which picture fits. A tarnished reputation has lost its shine, maybe slowly, maybe through the person's own doing. A besmirched one had something put on it. That's why the card's example has a man spreading false rumors, and why "my honor has been besmirched" always sounds to me like the speaker is about to name whoever did it.