I cannot speak to the others with authority, but this is incorrect.

BA Executive Club is the overall name for their frequent flyer program. That name includes both the reward aspect and the elite status (what you call loyalty) aspect.

Rewards within BAEC are based on Avios (essentially miles).

Elite levels within BAEC are based on tier points (a calculation that depends on flight length, class of travel, operating airline and fare class.

Iberia and BA are owned by the same parent company. Their frequent flyer programs are separate. Both call their reward currency Avios, but IB Avios are not identical to BA Avios. On a given flight, using BA Avios for an award ticket often incurs a higher fuel surcharge than booking the same award ticket using the same number of IB Avios. Also, earning elite status in BAEC and IBIP follow different rules.

Separately, airlines often operate two parallel programs: one for individual travelers, a second for small businesses. (For example, AA has AAdvantage for individuals and BusinessExtrAA for businesses.) A traveler who works for a small business can give both numbers and earn credit for both himself/herself and his/her employer. In theory a business of any size can enroll in the business program, but in practice large businesses can negotiate deals that are much more beneficial than the limited benefits of these programs and which preclude participating in the business program as well.