May not be entirely correct, but I do believe that historically "luck" and "happiness" were much closer to synonymous.
Recall that before many of modern technological and medical advances, one's entire status and well-being was attributed to how the gods favored them. In Europe, especially, the Catholic Church took a good amount of time to ingrain in people that if they were unhappy it was the will of god. As such it makes sense that if you are a "lucky" person, you are also a "happy" person. What person who is unhappy would ever consider themselves lucky? What person who is unlucky would ever consider themselves happy?
There are suggestions that "luck" was borrowed from the German "glück" as a gambling term some time before the 15th century. This would suggest that the word "luck" originated first and "happy" derived from it. In fact, all accounts point to the fact that in English and similar languages the evolution of the word "happy" began with "luck"
http://www.etymonline.com/index.php?search=happy&searchmode=none :
mid-14c., "lucky," from hap "chance, fortune;" sense of "very glad" first recorded late 14c. Ousted O.E. eadig (from ead "wealth, riches") and gesælig, which has become silly. O.E. bliðe "happy" survives as blithe. From Greek to Irish, a great majority of the European words for "happy" at first meant "lucky." An exception is Welsh, where the word used first meant "wise."
Note the Welsh exception, proving that this isn't necessarily a rule, although for the most part a common pattern in the 10th to 15th centuries for "luck" to be equated with "happiness".
This saying originated from a Middle English saying, round about 1545 A.D.
A coward verely neuer obteyned the loue of a faire lady.
[1545 R. Taverner tr. Erasmus' Adages (ed. 2) 10]
In 1614 A.D., this was refined to become:
Faint heart neuer wonne faire Lady.
[1614 W. Camden Remains concerning Britain (ed. 2) 306]
And later in 1754A.D., it was phrased in today's recognizable English:
Then, madam, we will not take your denial. ‥Have I not heard it said, that faint heart never won fair lady.
[1754 Richardson Grandison I. xvi.]
Thus is the origin of this saying.
These sayings were taken out of these books.
If you do not wish to browse through all those books for these few phrases, try this site
Best Answer
I think the origin of these phrases is from Shakespeare's The Merry Wives of Windsor, 1602:
As for which came first, lucky or charm, I found the charm variation earlier and not of American origin as The Phrase Finder has, but British. This is from The Cabinet Album, 1830 (date check):
And the lucky version I found three years later in The Port Admiral, by William Johnstoun N. Neale, 1833 (date check):
Since these two variations can be traced back to the same time period and the same country, I think it's safe to say they are related and that they both echo Shakespeare.
Edit: Heck, why not throw a pretty Ngram in for good measure: