We're talking about the batter-runner advancing to first base following a base-on-balls, so this changes things a little bit compared to the running lane requirements following a batted ball.
Yes, there is a different interpretation on this play for high school ball versus ASA! These two sanctioning bodies have published rulings to cover this. I really couldn't tell you about USSSA, NSA, etc. as I have never seen anything specific from them.
For High School softball (NFHS rules): There is an official ruling that the running lane rules DO apply when a batter receives ball four. When advancing to first base, a walked batter is bound to stay in the lane to avoid an interference call, just exactly the same as on a batted ball.
The ruling says that any throw to first here must be interpreted by the umpire to be part of "a legitimate play". An example would be a throw to a fielder at first base to prevent the batter-runner from rounding the bag (like you often see with a runner on third base).
As far as the catcher throwing AT the runner, as opposed to throwing TO a fielder on the bag, think of it this way: If the throw is in the direction and vicinity of the fielder, and the runner gets hit because she's out of the lane, that's interfence. If the throw is away from the fielder, such that the fielder had no reasonable chance of fielding the throw, or the throw is aimed at the runner, not the fielder, then it is NOT interference.
The throw must be a "quality throw"- that is, one that the fielder would have likely caught had it not hit the runner, before interference can be called.
There are a lot of variables and a lot of umpire judgment involved on this play (runner in the lane or out of the lane, quality throw or not a quality throw, catcher's intent to hit the runner or throw to the fielder). The call might go either way depending on any of those variables, but in high school softball you CAN get an interference call on this play.
For ASA: This is completely different!
ASA has a slightly different definition of "making a play" and that is the key reason why they interpret this differently. High school defines "a play" as "Any action by a fielder who is attempting to catch a batted or thrown ball". ASA defines "a play" as "An attempt by a defensive player to retire an offensive player".
For ASA, the action by the defense must have some reasonable chance to actually retire a player, or to record an out. Their interpretation is that with a walked batter advancing to first there is no "reasonable chance to record an out". Throwing down to stop a runner from rounding first base after a walk is NOT an opportunity to record an out.
Therefore, since there is no out to be had, there is no "play" to interfere with. If the runner is out of the lane and gets hits it's NOT interference. There's no penalty, it's a live ball...play on!