Did the 1897 St. Louis Browns really become the Cardinals? And did they then start up a new version of the St. Louis Brown, who eventually moved to Baltimore and became the Baltimore Orioles?
Yes and yes. Here's how it happened:
The Browns Stockings started in the American Association (the old "Beer and Whiskey League") in 1882. Chris von der Ahe, a German immigrant who owned a "beer garden", formed the team basically to sell intoxicants. He was sort of an early Charlie Finley/George Steinbrenner type. He hired future "Black Sox" owner and ballpark name-lender Charlie Commiskey to manage the club.
They were highly successful and won the AA crown from 1885-89 inclusive. They won baseball's original World Series, the Temple Cup, in 1886. The AA was folded into the NL in 1892 and St. Louis was one of the teams that survived. Commiskey jumped to the Reds and the Brown Stockings became a second-division team. They fell to 39-111 in 1898.
During the league wars, the NL started to swap shares of teams to improve overall league strength, until many owners owned shares of a few teams. The owner of the Brown Stockings, Frank Robison, also owned the Cleveland Spiders. He shifted all of the talented players including Cy Young to St. Louis in 1899. The Cardinals were dubbed the Perfectos for a short time. The Spiders became the worst team of all time and were forfeited to the league when four teams were lopped off after the season.
Other teams had similar practices: Pittsburgh pilfered from the Louisville club, that had an owner in common. That's how they got the great Honus Wagner.
The St. Louis club became known as the Cardinals around the turn of the century, although names were rarely official and were sometimes given to teams by the local media (That's why the "Stockings" was shortened to "Sox", in order to fit in headlines).
Meanwhile, Ban Johnson in 1894 took a rundown minor league, called the Western League. Charlie Commiskey, just fired by the Reds, took over the Sioux City club. By 1900, the league was the premier minor league in the country and Johnson renamed it the American League. In 1901 with the NL contracting by four clubs, they declared themselves a major-league and Commiskey shifted his franchise to Chicago's south side.
The Milwaukee Brewers club, one of the original Western-Leaguers, was shifted to St. Louis and the management selected the old-time name that had fallen into disuse, the Browns.
This was very popular in the AL: The early Baltimore Orioles took their name from the successful NL franchise from the 1890s. The Chicago franchise used White Sox in reference to the Cubs original nickname from 1871 to 1889 (then they became "Pop" Cap Anson's Colts when he managed them, then the Orphans when he retired). For a short time the Cubs actually barred the Sox from using Chicago in their name and they were officially just the White Sox for about two years.
The Red Sox stole their name from the Boston NL club in 1908. They had been known as the Americans, the Somersets, and the Pilgrims until 1907. In 1908 the Dovey brothers purchased the Boston NL franchise and decked them out in all white. The media dubbed the team the "Doves" in honor of the new owners. The Boston AL club quickly switched to red socks and dubbed themselves the Red Sox. In 1909 the NL club tried to switch back by changing their uniforms, but it was too late. The became known as the Rustlers and then the Braves (and briefly the Bees in the late '30s). The Red Sox became a passion to their fans; the Braves floundered for decades until they moved to Milwaukee and then Atlanta.
Fans knew the teams by a familiar name back then and if you could grap it, it was better than free bobbleheads for putting fannies in seats.
I hope that helps.