"Tenessyng handball, fott ball, stoil ball, and all manner other games out churchyard." - John Myrc, ca. 1450.
The "Stoil ball" referenced in this instruction to prohibit it from being played in the churchyard is likely one of the earliest mentions of a safe-haven game, the family best known today for baseball and cricket. A mention of a game called "creag" in 1300 and a "bittle-battle" even earlier have also been suggested as possible examples of the family, but as all we know about them are their names, it's not reasonable to say so with any certainty. (There's a proposed etymological connection between "creag" and "cricket," but this is probably spurious - most sources give a Dutch origin for the latter term, while "creag" is almost certainly taken from a Celtic word meaning "fun.") "Stoil ball," however, is clearly a game played with a ball and a "bar" - probably a bat. Later descriptions of games with clearly related names are also definitely part of the safe-haven family, though one should be cautious in projecting those back to the 15th century as our first written account of how the game is played is from three hundred years later. There are, however, a few visual sources which push the safe haven games back into the 1200s, so the suggestion that "creag" could have been such a game is entirely reasonable, and identifying "stoil ball" as part of the family seems fairly well-founded.
Baseball originated in the United States, and was derived from a variety of safe-haven games that had been played in North America from the colonial period. These mostly had a large number of common features almost certainly owing to the shared similarities of what had been imported from England. Many of the common elements are also seen in cricket, the world's most popular safe-haven game today, and the history of the cricket branch of the family is far clearer, with the professional sport developing in the 1620s and the first written codification of cricket rules being published about a century thereafter. A 1744 version of the cricket rules would see widespread adoption, and is the ancestor of the modern game. However, the early references suggest that until the 1590s, cricket and other safe-haven games were not perceived as being suitable for adults - at least among the gentry. (There's some evidence of stoolball having been primarily played by adult women, especially milkmaids.) This likely contributed to the lack of written sources about its precise rules.
Despite the complete lack of written descriptions of the rules of any safe-haven game prior to the 18th century, reconstructions have been attempted. Generally, the approach seems to be to take the common aspects of safe-haven games generally, without any of the quirks of a specific early game; this ensures that the players are not introducing an ahistorical twist, but since most safe-haven games have some distinctive feature in the rules, it misses out on some of the variety the historical games afforded. It is, however, commonly suggested that prior to the start of play, representatives from the two teams should discuss what precise rules they are using, which is likely to be a historically authentic aspect of any game between players from different towns.