Recalled -- Player is on the 40-man roster, has been in the majors before or at least in major-league spring training.
Purchased the Contract -- Player was not on the 40-man roster and thus had to be added before being promoted from the minors. The contract-purchasing is the act of putting the player on the 40.
Not sure about etc., but here goes.
Waivers -- Player passes through a system that allows other teams to take him and put them in a system, or on a roster, depending on the player's previous status. Most waivers work in the opposite order of the standings, by league. Thus, a team like the Yankees would have access to an American League player on waivers before the Cardinals or the Reds or any team in the NL.
Outright waivers -- Waivers for the purpose of removing a player from the 40-man roster.
Release waivers -- Waivers for the purpose of releasing a player.
Designated for Assignment -- One of the above operations is about to take place, but the team wanted time to see if there was another team interested in the player and a trade could be worked out. But in the mean time, the original team wants a free spot on the 40-man roster and needs to move this player off of the 40-man to make it happen.