The OP links to the Forbes Mar 18, 2017 article Ask Ethan: How Bright Is The Earth As Seen From The Moon? and @PM2Ring comments
FWIW, Dr Ethan Siegel is a qualified astrophysicist and an excellent communicator. You can generally trust his Ask Ethan articles. OTOH, they are pop-sci, and usually don't include the details of his calculations, or links to relevant papers.
So I'd say that it's likely we'll be able to generally confirm Siegel's unsupported assertions. As @PM2Ring points out, assertion-supporting isn't always part of popular science articles, but of course it's a cornerstone of Stack Exchange answers!
So let's have a go at some good old-fashioned Stack Exchange assertion-supporting. :-)
While someone operating a lunar rover (or an automatic one) would use headlights for safety, if you got lost at night and your power was batteries, you would need to nurse them for two weeks until sunrise.
Driving by Earthlight might be fun, necessary, and helpful for low-power autonomous rovers since reducing power for headlights might allow a longer range for a given battery mass.
Since the angular width of the illuminated part of the Earth's disk as seen from the Moon can be as much as 2° compared to the Sun's 0.5° shadows will look different, and that weird phenomenon where it's really hard to tell if something is relatively close or really far away that the Apollo astronauts have reported may not be so severe by Earthlight.
Therefore this is helpful to know and potentially important to know!
- Now large would the Earth be, and how bright, with comparisons, would it be? How brighter or dimmer would it be from the full moon, or a lamp?
- Can you read under the light of the Earth?
- And I know it would vary, so how bright would be at half full, or quarter full?
These are good questions and relevant to the experience of future Moon colonists, where dwellings and vehicles will have "Sun roofs" and "Moon roofs" but they'll be called "Earth roofs".
Since some folks have "cast shade" on this question of Earthlight, I'll post this answer as partial before a possible closure then continue to work on it.
There are relevant answers to several questions in Astronomy SE:
- How bright is the full Earth during the lunar midnight?
- How many times brighter would full-earth-shine appear to people standing on the moon, than full-moon-shine does to people on earth?
and here in Space SE there are answers we can draw from as well:
- Can you read under the light of the Earth?
@DavidHammen's answer to the 2nd question includes the following:
The full Earth as seen from the Moon is, on average, over 40 times brighter than is the full Moon as seen from the Earth. Reading a newspaper on the Moon at night under a full Earth would be a piece of cake compared to reading a newspaper on the Earth at night under a full Moon.
While there's no supporting calculation given, on matters of Space Exploration DH's opinions and assertions are pretty much "99 & 44/100% reliable" to borrow a phrase from space history and use it out of context for the purposes of levity. In reality, over the last 7 years and 5 months I have never seen a DH factoid I found fault with - and when I doubted them I followed up and was reassured.
So we can probably safely say that yes, you will often be able to read a newspaper on the Moon when illuminated only by Earthlight. Of course it may get pretty dim during new Moon and disappear when you're experiencing a lunar eclipse (penumbral or umbral) so it's probably not reliable.
- And I know it would vary, so how bright would be at half full, or quarter full?
I'll find an answer to this or calcul-estimate it myself. There is a large body of work on radiation from Earth and spacecraft heating in Earth orbit, but optical reflectivity way out at the Moon is a different thing. We have to use an average albedo -- weather causes the fraction of white cloud cover to vary a lot, so weather on Earth will affect how easy it is to read a newspaper on the Moon by Earthlight! Also, oceans have a specular-like component and a diffuse-like component, depending on the local wind history. That effect is in fact used to measure wind speeds by reflection of GPS signals!
- This answer to How can the CYGNSS spacecrafts (actually) measure ocean roughness?
- Apparent magnitude of a spherical body with specular, rather than diffuse reflectivity? How bright were Sputnik 1 and Vanguard 2?
I will draw from a calculation I did for the brightness of Venus or other astronomical bodies as a function of phase angle and apply it to this geometry. See for example
- Answers to Calculating the apparent magnitude of a satellite
- This answer to Venus' magnitude during inferior conjunction but I'm looking for an independent calculation, stay tuned...
- This answer to Does anybody know how to solve this Earth-Venus-Sun problem?
- Now large would the Earth be, and how bright, with comparisons, would it be? How brighter or dimmer would it be from the full moon, or a lamp?
I'll find answers to this as well... Stay tuned!