signed as free agent?

A free agent player is under obligation to no team in the league and his last team holds no rights to compensation from whichever team obtains him -- with one exception. If the free agent player was offered arbitration and declined it, and currently qualifies as either a Class A or Class B ranked free agent, his former team will gain one or more draft picks in the following Rule 4 (amateur) draft, and sometimes (depending on the player's ranking, mostly) one of those picks will be forfeited from the team that signs him.

That's FAs who have already played professionally. Amateur free agents are just that -- amateur players, undrafted (or drafted and since released), who can be signed by any team. Mostly these tend to be from other countries, as the draft only covers US citizens, but some American kids turn up here and there and get signed without going through the draft.

When a player is a free agent, they are unemployed, so when one is signed as a free agent, that means that they have a contract and will play for whatever team offered him a job.

It can, or it could be their current contract ran out with a team, and they joined the team via free agency. Basically, it means they are joining a team because they don't have a contract with another team, be it they were undrafted or they are not currently emplyed by another team.

Who says "for university?"