Chatter always seems to center around the latter, but it's the former that's landed them in the NFL.
"I know he got a lot of scrutiny [from] people thinking he couldn't throw," said Arizona Cardinals rookie Murray of his Baltimore Ravens counterpart Jackson. "Then he goes out there and proves it, so people say things about everybody."
When Murray and Jackson stand on opposite sidelines of M&T Bank Stadium in Baltimore on Sunday (1 p.m. ET, Fox), they'll be representing the evolution of the dual-threat quarterback. They're not running quarterbacks. They're quarterbacks who can run. The distinction is important.
"As a quarterback, when you run, that sometimes gets overhyped and your throwing gets shadowed a little bit," said Cardinals backup quarterback Brett Hundley, a five-year veteran who ran often in high school and college. "Then you don't realize how good he's throwing the ball because you're seeing him running the ball, where most quarterbacks throw it away or throw the checkdown or something like that.
"We just take off and run. So, it's like, 'Oh, he's a runner.' But, instead, it's just we're doing different things when nothing's open so I think that's the biggest difference."
Hundley would know -- he's backed up the likes of Aaron Rodgers in Green Bay and Russell Wilson in Seattle before supporting Murray in Arizona. He considers all three to be quarterbacks who can run, not the other way around.
So why the labels on Jackson and Murray? Jackson had back-to-back 3,000-yard-passing/1,500-yard-rushing seasons at Louisville, earning him a Heisman Trophy. Murray closed out his collegiate career at Oklahoma with a 4,000-yard-passing/1,000-yard-rushing season that earned him a Heisman. Having success as a runner in college starts the stigma. Wilson, Robert Griffin III, Donovan McNabb and Randall Cunningham, among others, were typecast when they arrived in the NFL.
"It's gone on forever, that's always been the thought," Cardinals coach Kliff Kingsbury said. "If somebody comes along that proves that wrong, then a lot of people are wrong and people don't like being wrong. I think that's where that stands currently."
Murray and Jackson shined with their arms in Week 1. Jackson threw for 324 yards and five touchdowns. Murray for 308, two touchdowns and an interception. Neither ran a lot -- three times each -- as Murray gained 13 yards and Jackson six.
After his breakout game on Sunday -- a 59-10 blowout of the Miami Dolphins -- Jackson quipped to reporters that it was "not bad for a running back," a clear shot at his critics who thought he wouldn't make it as a quarterback in the NFL.
"The way he said it was really great," Ravens coach John Harbaugh said. "He was very humble. He kind of said it with a laugh with his eyes kind of down but it's probably some point he wanted to make for all those out there that want to categorize people in certain ways."
All those labels have been disappointing to Harbaugh but Jackson has enjoyed the stir created by his comment -- one rooted in proving people wrong.
"You could say that," Jackson said.
Murray knows the feeling. He heard the whispers when he transferred from Texas A&M to Oklahoma after his freshman season in 2015. The rap on him back then was that he was a short quarterback who couldn't throw.
"Bad combination," he said wryly.
Murray "for sure" felt like he had to prove himself, which he did last season at Oklahoma when he threw for 4,361 yards while running for 1,001. He won the Heisman and was the first pick in April's draft.
His rushing total fueled the chatter pre-draft that he was a running quarterback. Last Sunday began to put that notion to rest and every Sunday to come can reinforce it.
"The biggest thing is understanding that we can throw the damn ball, too," Hundley said. "Now you see a lot of quarterbacks coming through the pipeline that are both runners and passers."
Neither Murray nor Jackson will shy away from having the ability to make plays with their feet. And neither will shy away from using them.
"The ability to run is definitely changing the game," Murray said.
It's a weapon that can be tough for defenses to scheme against, Harbaugh said.
Cardinals rookie defensive lineman Zach Allen saw that first-hand when he played at Boston College against Jackson at Louisville. The Eagles schemed to prevent Jackson from running during Allen's junior season. It didn't matter. Jackson scored two touchdowns on 73 yards rushing while throwing for 354 yards and a touchdown.
"They're both fast," Allen said of Jackson and Murray, who he's faced since rookie minicamp. He called both "elite" passers but said their running skills force defenders to think twice.
"You really have to be on point and focus every play because they can do it in any way," Allen said. "It just makes our job harder, but also it's a challenge and it's a lot of fun. Do you make a conscious effort not to overcommit against guys like that?
"They can run by anybody."
Still, running as a quarterback in the NFL is a risk. Too much is at stake for any given franchise, especially when quarterbacks are receiving megacontracts. But, not surprisingly, it's the quarterbacks who think it's sustainable to run.
Kingsbury? Not so much.
"It isn't until somebody proves that it is," he said.
The key, Seahawks quarterback Russell Wilson said, is to be smart.
"It's getting down, understanding when to get out of bounds, not taking unnecessary hits, especially when you have the football in your hands," Wilson said.
"It's one thing to ... sit in the pocket and you got to get hit or whatever. That may happen. Those things are going to happen, for sure. But it's another thing to run. You don't want to get smacked when you're running down the field. I think just having the intelligence and understanding when to get down, how to slide, all those things is critical."
Murray doesn't like getting hit. Once he gets his yards, he's planning on getting down.
But there's a difference between Murray and some other quarterbacks. He doesn't run unnecessarily, said Cardinals quarterbacks coach Tom Clements. That's why Clements doesn't think Murray is similar to Michael Vick, a popular comparison. Clements believes Vick was more keen on running earlier in his career before developing into a better passer.
The better comparison for Murray, Clements said, is Chiefs quarterback and former Kingsbury recruit Patrick Mahomes.
"They're gonna sit in the pocket and look to throw," Clements said. "Then, if it breaks down and they have to create something, they can do it."
Murray wants to "play this game for a long time." To do that, though, he needs to stay healthy. To stay healthy, he needs to be smart. And it helps that he's fleet of foot.
"Guys are a lot bigger in the NFL than they were in college, faster and stronger," he said. "At the same time, I'm fast so I can get away."
That's one way to make running work in his favor. But that doesn't always work, Griffin III said.
"Guys are getting hurt as much outside the pocket as they are inside the pocket," he said. "Guys are getting demolished inside the pocket. You see those injuries happen a lot. There is no way to really avoid that. But if you're a guy who is mobile, you have to use that ability."
Quarterbacks who can run aren't new to football, Harbaugh said, but they may be faster and more athletic. And they may be able to throw better now, limiting their need to run.
Murray and Jackson are examples of both.
"We're throwers first," Hundley said. "We play quarterback. But since we run a lot, we get stigmatized as runners and so sitting back in the pocket and not getting hit, I'm fine with that."
ESPN NFL reporters Jamison Hensley and Brady Henderson contributed.