Dolphins quarterback Matt Moore is hit by Jets inside linebacker David Harris in New York’s 24-6 victory last month. (Bill Kostroun/ASSOCIATED PRESS)

When it comes to the Miami Dolphins, loyalties here are sharply divided. Everybody around town wants the Dolphins to win. What’s in dispute is the definition of “win.”

A few more victories like Sunday’s unexpected 31-3 drubbing of the Kansas City Chiefs would put the Dolphins, now 1-7, on the road back to respectability. But such a turnaround would expel them from the race for Stanford quarterback Andrew Luck, the presumed draft prize for this year’s worst NFL team.

Though fans of every franchise in the ignominious competition to tally the fewest victories covet the talented Cardinal quarterback, Dolphins supporters may possess a deeper, more profound yearning. It’s a longing that loyalists of the Washington Redskins, who play the Dolphins on Sunday at Sun Life Stadium, perhaps can appreciate more than anyone.

In the 12 years since Hall of Famer Dan Marino retired after the 1999 season, the Dolphins have used 15 starters at quarterback, including two this season, and won one playoff game (back in 2000). In the same period, the Redskins have used 13, including two this season, and won one playoff game.

For some, the chance to draft a blue-chip prospect that could help restore the franchise’s once-lofty tradition trumps any interest in salvaging this season. In an informal poll taken recently by Miami Herald columnist Greg Cote, nearly 70 percent of respondents supported the idea that continuing to lose games this season would be worthwhile.

The idea truly rankles players and coaches.

“Nobody likes hearing it,” Dolphins Coach Tony Sparano said during a quiet moment last week. “There are a bunch of competitors in that locker room. I find it offensive on their behalf because I know how hard those guys in that locker room work. I know how hard the people in this building work. . . . There’s only one agenda here, and that’s to win games. . . . Anything else is ridiculous.”

In Miami’s last home game Oct. 23, the Dolphins were booed after their first series and many in the less-than-sellout crowd cheered for — and sported the jersey of — Denver Broncos quarterback Tim Tebow. The former Florida Gator led a last-minute comeback 18-15 victory that dropped the Dolphins to 0-6.

“It does affect players to some degree,” Dolphins defensive coordinator Mike Nolan said. “You want the home-field advantage. . . . The last home game, it felt like an away game a little bit right there.”

Prior to last Sunday’s victory, the Dolphins had lost 10 straight dating back to Dec. 12, 2010, when Miami defeated the New York Jets on the road. Sparano has faced questions about his job status nearly every week since. But in Sunday’s dominating victory, the Dolphins played well offensively and defensively and, for the first time in two weeks, did not give up a late lead.

Quarterback Matt Moore, who assumed the starting job after Chad Henne suffered a shoulder injury in the season’s fourth game, struggled in his first three outings (as has the Redskins’ John Beck, a former Dolphin who replaced Rex Grossman in the fourth quarter of Washington’s Oct. 16 loss to the Eagles and has kept the job since). But on Sunday, Moore threw for 244 yards and three touchdowns in a performance that was close to masterful.

“It was absolutely great to get a little reward for all the work we’ve put in, just to have that feeling of getting a win under your belt,” he told reporters after the game. “I think guys now understand that, ‘Yeah, we can do this,’ and they’re hungry and we’re going to enjoy this one and get ready for Washington.”

Despite Moore’s stellar play against Kansas City, few expect him to bring back memories of Marino this Sunday or any other. Undrafted out of Oregon State in 2005, he has thrown for 20 touchdowns and 21 interceptions and earned a 75.4 rating in five NFL seasons, four with the Carolina Panthers. The question in Miami is not whether he will lose the job next year, but who will take it from him.

Southern Cal’s Matt Barkley and Oklahoma’s Landry Jones also are pegged as likely first-round picks, but neither is considered in Luck’s class.

Like the Redskins, Miami has churned through quarterbacks for more than a decade. Only one has had a top-notch season — Chad Pennington earned a 97.4 rating while the Dolphins won 11 games during the 2008 regular season. Since Marino’s departure, the team’s primary starter has failed to top a 76.0 quarterback rating (such a rating would be 28th overall among quarterbacks this season) in eight of 11 seasons.

The Dolphins have recycled a handful of signal-callers discarded by the Redskins, and vice versa. Beck started four games for Miami in 2007. Former Redskins Gus Frerotte, Sage Rosenfels and Trent Green also have started for the Dolphins in recent seasons.

None excelled.

Despite this season’s struggles, players have offered their public support for Sparano, who recently put his house in South Florida on the market, saying it was only in order to move into a smaller property close to the beach. The team has earned a reputation for playing hard, even in defeat, a trait that suggests what Sparano has insisted: that the season is far from over.

Which might not be good news for the team’s die-hard supporters.

“We don’t want to disappoint fans,” said defensive end Randy Starks. “We’re frustrated just like them. At the same time, we can’t let it distract us. We have to go out there and turn this around. We’re like three or four games away from it being a different season.”