Practical formula/method to predict the necessity of termination resistor
The scope input (standard BNC type) is either 50 ohms or 1M ohm. The user selects the type as part of the input setup, or the scope may auto-detect the kind of probe that is attached.
For scopes, 'practically' then:

Passive probes use 1M
Active probes use 50 ohm
Direct RF cabling uses 50 ohm

For the active probe and direct cases, the scope is terminated in the cable's characteristic impedance to absorb any reflections and maintain good signal fidelity (integrity) at the input. Without that termination, the waveform will be severely distorted and render the measurement unusable.
If 'some non-engineer researchers' (easy there, don't be an elitist) were using a 50 ohm source (like a signal generator), and were measuring the waveform with the 50 ohm termination switch on, it would have worked just fine. In fact if they were not using the high-impedance probe it's the right way to do it. The downside would be that the signal would be loaded down so the voltage swing would be lower (for a 50 ohm source, by 50%).

So you'll hear talk about 'transmission lines' and 'characteristic impedance'. What this means is that a given cable has, per unit length, a unit inductance and unit capacitance. The cable's net impedance is, roughly speaking:

\$Z_0 = \sqrt{\dfrac{L}{C}}\$

where L and C are the unit inductance and capacitance, respectively.
What determines these L and C values? The unit inductance is mostly about how thick the coax wire is, and unit capacitance is the dielectric thickness and its material type that in turn determines its permittivity.
More here: How is xΩ impedance cable defined?
Why 50 ohm? This cable impedance was settled upon as a compromise in the early days (1930s) of cable development for radio transmitters. More here: https://www.microwaves101.com/encyclopedias/why-fifty-ohms
Today, 50 ohm is the impedance of common cables like RG58a/u and remains the main choice for microwave and high-speed signal link work like PCI Express, SATA, USB3.0 and other fast serial interfaces.
75 ohm is the the other widely-adopted choice, used mainly for UHF/VHF and cable television. It is preferred for that use owing to its lower capacitance and reduced signal loss. More here: http://cablesondemandblog.com/wordpress1/2014/03/06/whats-the-difference-between-50-ohm-and-75-ohm-coaxial-cable/
To use 75 ohm directly on a 50 ohm scope, an impedance matching network is used.