When people generally think of BYU the first thought isn't the accounting program...or Jimmer...or any of the various QBs (Young, Detmer, Wilson etc). It's Mormons.
When I was at the Air Force Academy, we had a noon meal formation just about every day. On Fridays before a college football weekend, the entire cadet wing would yell something out at the end - "Beat the Utes" or "Beat Wyoming" or the like. Pretty much always "beat the _____" with the school name or identity thrown in.
The only exceptions were occasionally Navy ("beat the squids") and always BYU ("beat the Mormons").
For most outside Utah, trying to define BYU beyond the Mormon faith (and I'm using Mormon here purposely instead of LDS) is like the situation where you're trying to point someone out while avoiding the obvious descriptor (only black guy in a group of otherwise white folks, or the guy who is 380 pounds) -
"yeah, you know Darrell, you met him at my birthday. He was the tall guy in the red shirt. No not a dark red shirt, the light one. Darrell's the one with the darker hair, he was there with his kid. Uh, forget it, he was the only 380 pound black guy there."