Windhund, der -
1. großer Hund mit langem, schmalem Körper, schmalem Kopf, langem, kräftigem Schwanz und seidigem [langhaarigem] Fell
2. leichtsinniger, oberflächlicher, unzuverlässiger Mann (umgangssprachlich abwertend)
Synonyme - Leichtfuß, Luftikushttps://www.duden.de/rechtschreibung/Windhund
Leichtfuß, der -
leichtfertiger, leichtsinniger Mensch
Synonyme - (umgangssprachlich abwertend) Luftikus, Windhund; (umgangssprachlich scherzhaft) lockerer Zeisig; (salopp, oft scherzhaft) lockerer Vogel; (bayrisch, österreichisch umgangssprachlich) Hallodri; (ostmitteldeutsch) Schlenkerich; (veraltend abwertend) Windbeutel; (veraltend scherzhaft) Bruder Lustig/Leichtfuß/Liederlich; (veraltet) Holdriohttps://www.duden.de/rechtschreibung/Leichtfusz
Luftikus, der -
leichtsinniger, oberflächlicher, wenig zuverlässiger Mann
Synonyme - (umgangssprachlich abwertend) Windhund; (umgangssprachlich scherzhaft) Leichtfuß, lockerer Zeisig; (salopp, oft scherzhaft) lockerer Vogel; (bayrisch, österreichisch umgangssprachlich) Hallodri; (ostmitteldeutsch) Schlenkerich; (veraltend abwertend) Windbeutel; (veraltend scherzhaft) Bruder Lustig/Leichtfuß/Liederlich; (veraltet) Holdriohttps://www.duden.de/rechtschreibung/Luftikus
Windhund - 1. greyhound;
Afghanischer Windhund - Afghan hound
2. (ugs. abwertend) careless, unreliable sort (coll.)
Windhund - a: (Hund) greyhound; (= Afghanischer Windhund) Afghan (hound) b: (fig pej) rake
Windhund - greyhound; (fig.) harum-scarum
So far both the definitions and the translations seem to be pretty much all over the map. I've collected a few more sources; I don't know if anyone still has a Wahrig anywhere and if it might offer yet another angle on the German side.
I tried to just brainstorm on some of the aspects that seem to come into play. I don't think it actually got me anywhere, but for what it's worth ...
'Rake' is an old-fashioned (~18th-c.) term for a man about town, a Don Juan, Casablanca, roué, libertine; that is, a man who has sex with many women, but in a romantic or attractive way.
'Reprobate' is more just a scoundrel, bad guy, rogue, villain, crook; a man who cannot be trusted and who is inclined to break the law. 'Lowlife' and 'nogoodnik' are much more modern in a similar range, though somewhat milder. Stronger terms might be scumball, slimeball, jerk, ...
'Weasel' might also be in a similar general group, though to me it emphasizes more someone who weasels out of commitments, who is likely to betray or deceive others. Related adjectives might be unscrupulous, slippery, two-faced, double-dealing, or similar nouns might be a rat or a snake.
'Adventurer' in more recent usage may be more often positive: an adventurous person, explorer, discoverer. In more old-fashioned usage it was more negative, from a daredevil to a fortune hunter.
'Flighty' to my ears means fickle, frivolous, inconstant, and a little dim-witted or featherbrained, especially in an old-fashioned way that used to be thought of as typically feminine. (Cf. the aria 'La donna è mobile / qual piuma al vento.') Many actresses used to play this kind of wide-eyed dumb blonde role, from Marilyn Monroe to Goldie Hawn.
For a man, you could say he was irresolute or indecisive, naive, shallow, a lightweight, someone who wavers or dilly-dallies, who trifles with or makes light of serious subjects, who is wishy-washy, namby-pamby, scatterbrained, foolhardy, heedless, a dreamer, woolgatherer, slacker ... But none of those are quite the same; it may be that English has traditionally associated that kind of personality more with femininity.
'Cavalier' (adj.) could be close, but is probably more often used to describe a particular action than an entire person's character. To me it connotes being irresponsible in the sense of not taking something seriously enough, taking it too lightly or carelessly. You can be cavalier with, or have a cavalier disregard for, the truth, the law, or someone's feelings; you can brush something off or dismiss it in a cavalier or facile way.
'Harum-scarum' is old-fashioned for just a rascal, scamp, mischief-maker; to me it implies a certain boyishness and wouldn't be as typical of a grown man.
In many cases, the adjectives that correspond to the Duden definition just don't seem to have an obvious noun form that would typically be used of a man.
leichtsinnig - careless, reckless, negligent
oberflächlich - superficial, shallow
unzuverlässig - unreliable
The Oxford solution with a couple of adjectives may be as close as we can get, unless someone else has a flash of inspiration.