How as light intensity measured or determined around 1900? For example, when it was determined that the photoelectric effect was independent of the intensity of the light beam, but rather that it depended on the wavelength, how was the intensity measured?
There was no need to measure the intensity directly to determine that the photoeffect is independent of it. One could vary intensity by moving the metal plate closer to or further away from the source. How intensity falls with the distance is well-known. This is what Lenard did in his 1902 experiments that prompted Einstein's 1905 work.
If there was a need to measure intensity, Lenard's preferred method was to use phosphorescent materials to make the ultraviolet (or cathode, now X) rays visible, rather than materials that get heated by the light. However, absolute units of intensity were poorly standardized at the time even for visible light. In Germany, Hefnerkerze (Hefner lamp) developed by Hefner in 1884, was used for the purpose. One hefner was defined as the horizontal intensity of the light from such a lamp, and is approximately 0.903 candela. Candelas were only standardized in 1948.
Here is Lenard's original paper Ueber die lichtelektrische Wirkung (About the photoelectric effect, 1902). In English, he recounts his methodology in the Nobel lecture On Cathode Rays (1906):
"It must be noted that the rays are not directly visible; it would be useless to put one’s eye to the window, as this organ is not receptive to cathode rays. On the other hand, materials that are capable of becoming luminous without heat, phosphorescent materials as they are called, are suitable for making the rays visible. It is best to use sheets of paper coated with such materials, e.g. a certain ketone, platinum cyanide, or an alkaline-earth phosphor and to hold them as a screen against the rays. If the screen glows, it indicates that it has been hit by the rays. The rays can also be photographed directly.
These are the same methods that are used to make visible ultraviolet light, at that time the only known example of such demonstrable invisible radiation. When we use the phosphorescent screen, we find it glowing brightly close to the window; as the distance from the window increases, the intensity of the rays progressively diminishes until at a distance of about 8 cm the screen remains quite dark.
[...] I have also found that the velocity is independent of the ultraviolet light intensity (M), and thus concluded that the energy at escape does not come from the light at all, but from the interior of the particular atom. The light only has an initiating action, rather like that of the fuse in firing a loaded gun. I find this conclusion important since from it we learn that not only the atoms of radium - the properties of which were just beginning to be discerned in more detail at that time - contain reserves of energy, but also the atoms of the other elements; these too are capable of emitting radiation and in doing so perhaps completely break down, corresponding to the disintegration and roughening of the substances in ultraviolet light."
The last paragraph sketches Lenard's (classical) explanation of photoeffect, the "triggering hypothesis", which was largely accepted until 1911, contrary to the now popular opinion, see Were there serious attempts to model the photoelectric effect classically? It was disprioved by Lenard himself, ionization in gases by ultraviolet light did not produce strong ionization unaccompanied by absorption of the light. For a historical commentary see Philipp Lenard and the Photoelectric Effect, 1889-1911, by Wheaton:
"It is only since 1911 that the photoeffect has been interpreted as a transformation of light energy into electron kinetic energy. In 1902 Lenard concluded that the effect is a resonance phenomenon, but one in which the light contributes no energy to electrons. The light only selects which electrons are to be ejected from an atom. The velocity of a released electron is predetermined within the atom. Consequently a study of the velocity distribution of photoelectrons promised to shed light on atomic constitution."
[...] For nine years the triggering hypothesis was the accepted explanation of the photoelectric effect. It enabled physicists to avoid the serious difficulties which obstructed formulation of a consistent mechanism for the absorption of classical electromagnetic waves. The light quantum proposed by Einstein to answer more general issues incidentally offered another explanation of the photoeffect. But other physicists preferred the triggering hypothesis."