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The maximum distance at which a phone can still make calls and send texts varies depending on the type of tower and transmitter, but an airplane would have to be no more than 10,000 feet in the air for any cell phones on board to still have a signal, according to Bill Rojas, director of telecom research at IDC Asia Pacific.

https://www.cnn.com/2014/04/14/tech/mobile/phones-in-flight/index.html#:~:text=Despite%20the%20urgings%20of%20flight,of%20feet%20into%20the%20air.

How were passengers able to make cell phone calls? Did all of them use airphones?

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No, I believe very few of them used satellite phones.
An article on the New York Times from 14-09-2001 states:

According to industry experts, it is possible to use cell phones with varying success during the ascent and descent of commercial airline flights, although the difficulty of maintaining a signal appears to increase as planes gain altitude. Some older phones, which have stronger transmitters and operate on analog networks, can be used at a maximum altitude of 10 miles, while phones on newer digital systems can work at altitudes of 5 to 6 miles. A typical airline cruising altitude would be 35,000 feet, or about 6.6 miles.

Now, this theory was valid then (during 9/11) but not for modern aircraft like the A320, 737s, etc. Why is that so?

Jon Brittingham, a technical pilot in the Airbus A319/320/321 program, explained the causes to me in greater detail. Older aircraft, such as the McDonnell Douglas MD-90, don’t have the same electronic-systems shielding to protect cabin equipment from third-party signals that more modern airliners, such as the current Airbuses, have. In fact, this shielding is most dense around the front of the cabin—thus confirming my pilot friends’ agreement about poor reception within the cockpit. (This same fact can also cause cell reception in first-class seats to be the worst in the plane, when parked by the gate.)

In modern day aircraft, signal reception is poor even when on ground (but inside the aircraft), it becomes worse on takeoff and negligible while climbing and on cruise.

According to me two factors come in hand, Older cellphones had stronger signals, previous generation aircraft (like the B757 in 9/11) might have lacked proper shielding to prevent 3rd party EM signals.

You can refer to the following articles:

Fun Fact: I've observed mobile phone signals are terrible and mostly do not work on boarding a 787. It turns out that the 787, while fabricated from composite materials, uses a wire rebar for structural integrity. That rebar apparently interferes with cellular signals on the plane.