I'm not so confident as you that there is a single unimpeachable definition of an LCC but everything I'm finding makes it clear that fare price is intrinsic to the idea of an LCC and ULCC.

You must be seeing some other source that's more definitive than I'm able to find.

Low-cost carriers refer to the five U.S. airlines with the most enplanements in 2010 that embrace a business model that emphasizes maximum load factors coupled with streamlined passenger and aircraft operations (Southwest, Jet Blue, Frontier, Spirit, and Allegiant).

https://www.bts.gov/archive/publications/passenger_travel/chapter4
https://www.bts.gov/archive/publications/passenger_travel/chapter4/figure4-5

For the traveler, these airlines may more commonly be referred to as budget airlines, discount airlines, low-fare airlines or no-frills airlines. These terms refer to the fact that the consumer typically spends less money to fly on low-cost carriers. Low-cost airlines specialize in keeping the costs down, minimizing their operating costs by having single aircraft fleets, and without some of the more traditional amenities (like in-flight meals) included in the fare,  meaning they can offer lower fares due to their lower cost of operating – essentially these cost-savings are being passed on to the traveler and for LCCs, the trade off is they stimulate more passengers to travel due to lower fare[...]

https://www.oag.com/blog/what-are-low-cost-carriers-aviation

A second observation: although LCCs have pursued a number of different business models, common to all is a laser-focus on the customer: identifying what prospective airline passengers value—that is, what they are willing to pay for—and then offering them products to meet that demand.  

Third:  in a cyclical industry subject to surges in the cost of fuel, not to mention crises such as terrorist attacks and health scares, LCCs have understood that maintaining a competitive advantage requires a relentless drive to cut costs, expand revenues, and maximize efficiency.  

https://www.icao.int/low-cost-carriers-lccs

A low-cost carrier (LCC) is an airline that offers budget-friendly airfares by operating with a lean, cost-focused business model. These carriers minimize overhead by simplifying services, standardizing fleets, and flying point-to-point routes—often using secondary airports. The result: lower base fares, but limited included amenities.

Key Characteristics of Low-Cost Carriers:
Low Base Fares: The ticket price usually covers only transportation; extras like checked baggage, seat selection, meals, and boarding priority come with additional fees.

https://www.business-class.com/glossary/low-cost-carrier

A low-cost carrier (LCC) or low-cost airline, also called a budget or discount carrier or airline, is an airline that is operated with an emphasis on minimizing operating costs. It sacrifices certain traditional airline luxuries for cheaper fares. To make up for revenue lost in decreased ticket prices, the airline may charge extra fees, such as for carry-on baggage.

The term originated within the airline industry referring to airlines with a lower operating cost structure than their competitors. The term is often applied to any carrier with low ticket prices and limited services regardless of their operating models.

https://en.wikipedia.org/wiki/Low-cost_carrier