Neither proposal is being pushed or sanctioned by any formal organization or government agency; both, like the monorail in its earliest days, are the work of individual transit supporters.
The first line, part of a theoretical "Seattle subway" system, would start at Westlake and travel along the original monorail route through Seattle Center, Queen Anne, and Interbay, terminating at 15th and Market in Ballard before eventually heading up to Crown Hill and Lake City. Because light rail can go underground, at-grade, and on elevated tracks, it could theoretically cross the Ship Canal in a tunnel, instead of a second Ballard Bridge.
The second proposal, called the "Ballard Spur," would start at Brooklyn Ave. NE in the U District, linking to the funded North Link light rail line at Sound Transit's Brooklyn station. From there, the train would travel along 45th St. through Wallingford, Fremont, and into Ballard.
In my view, there are good and bad elements to both routes. Although building rail from Westlake to the Ballard Bridge would be relatively inexpensive, tunneling under the Ship Canal would be costly---the tunnel under the Montlake Cut is one of the most expensive elements of Sound Transit's North Link rail line. Additionally, I remain skeptical that Ballard is poorly served by buses now. In 2012, a new RapidRide bus-rapid transit route will open between downtown and Ballard, following exactly the same route that the light-rail plan would follow. On the other hand, support for light rail between downtown and Ballard remains high. And dense downtown Ballard is exactly the kind of community transit supporters are referring to when they talk about "transit-oriented development."
The 45th Ave. route presents its own unique challenges. It's unclear whether the route would be on the surface or below grade; tunneling, obviously, would be much more expensive, but 45th St. is already extremely congested and narrow (two travel lanes, one turning lane, and two parking lanes); adding rail tracks would require removing one or both lanes of parking, a proposal that would almost certainly prompt an uproar from the neighborhood. On the other hand, Seattle suffers from a major shortage of east-west transit service. The only route that carries riders between the U District and Ballard, the 44, is crowded, slow, and frequently stuck in traffic. Like Martin Luther King, Jr. Way in Southeast Seattle, 45th is a corridor where grade-separated rail just makes sense.
Perhaps the biggest hurdle for either proposal is that voters will have to approve any rail extension---and voters, as the recent car-tab vote shows, aren't enthusiastic right now about big spends for "frills" like transit when basic road maintenance is going unfunded.