The author dismisses cubemaps pretty quickly, but imo it’s the simplest solution & it’s what I did when rendering dynamic gas giants on my own personal project a number of years back* . Using a cubemap doesn’t result in a 6x increase in memory usage, you’re just splitting the texture size from one large rectangular face into 6 smaller rectangular faces, but the total texture detail is the same. The nice part about a cubemap is you don’t have to worry about pole pinching at all + you can use a 3 or 4 dimensional noise function to easily create a seamless flow field for texture animation/distortion.
If you want to avoid weird seam artifacts, using 2 hemispheres of stereographic projection is probably better still. Each hemisphere projects to a disk but you can just fill out the texture to the boundaries of a square, duplicating a bit of each hemisphere into the corners of the other's texture (or you could leave those parts blank if you want). There's a 1:2 difference in scale from the center to the edge of each disk, so you could argue this is slightly wasteful of pixels for a given minimum required level of detail, but the projection is conformal so it's considerably less tricky to figure out how to sample it to decide the color for destination pixels drawn at steep perspective, and the stereographic projection is very cheap to compute in both directions (1 division + some additions and multiplications per projected point), even cheaper than the gnomonic projection used for a cubemap.
If you want something conformal that has less scale variation and wastes fewer corner pixels than a pair of stereographically projected hemispheres and is still not too conceptually tricky, you can use a pair of slightly overlapping Mercator projections, at right angles to each-other, covering the sphere like the two pieces of leather covering a baseball. Each one can have a rectangular texture. There are some NOAA papers suggesting this approach for the grids for solving differential equations needed in weather simulation of the Earth.
The most pixel-efficient projection I know starts by breaking the sphere into an octahedron, then taking each octant to be covered in a grid of hexagonal pixels, using "spherical area coordinates" in each octant to determine the grid. Each octant can then be represented in an ordinary square-pixel image by a half square ("45–45–90 right triangle"), so the result is something like this <https://observablehq.com/@jrus/sac-quincuncial> with a hexagon grid like <https://observablehq.com/@jrus/sphere-resample> (scroll a few examples down from the top of the page). But figuring out the details about how to sample the texture when you need to cross edge boundaries, etc., makes using this quite a bit more fiddly than the 2 stereographic projection version. And there will be some seam artifacts.
And if you want to further minimise distortion, you could group the triangular faces of an icosahedron into 10 rhombuses, each covered by one square texture.
It's more math, though, and usually not worth it unless you're already planning to subdivide the surface further for some other reason.
I’ve since done a bunch more on planetary cloud rendering, need to do a proper write up, but it’s a combination of volumetric noise, flow fields & atmospheric scattering.
* https://www.junkship.net/News/2016/06/09/jupiter-jazz