Let's try with, hated for good reasons, boolean workflow, mainly because I think the tutorial you linked uses booleans. Though there the holes aren't distributed evenly:

The above could be achieved by making a cone, and adding two array modifiers, but we can do better, like Chris, so let's try but with boolean modifier.
The advantage of boolean is that you don't have to go through a puzzle on proper topology, so this solution should be more versatile (e.g. when you want to replace the cones with cylinders as on one of your reference images). The disadvantages are, however, plenty:
- the resulting topology may be very heavy (a lot of triangles)
- booleans are often buggy and cause issues with animations due to inconsistency
- there are also problems with smooth shading and resolving them can be quite tricky (normal data transfer from the original uncut sphere)
Add an Ico Sphere (because that generates equally spaced triangles), then add a cone, edit it, using Increment Snapping
(hold Ctrl or click the magnet button) move the cone so that the tip is at the world (sphere) center. I also move the base up because I find it more natural to look an an object from the top, but that's just my preference. Then I rotate the cone to make it roughly fit a triangle (keep in mind in this situation you have to rotate along green Y axis, not red X axis, due to how the triangles are distributed). If you readjust the distance of the base of the cone to the center, remember to use G, Z but in local coordinate space, ignoring the rotation of the object. 

Let's name this cone Cone.A, and duplicate it, and name the duplicate Cone.B, let's also Hide Cone.B for now. Likewise rename the Ico Sphere to Ico.A, and create a duplicate Ico.B, hide the latter.
Select Cone.A, then Ico.A, Ctrl+P, O.
Edit the Ico.A, select the triangle overlapped by the cone, Shift+S, U:

Now in object mode, with the Cone.A selected, press F3 and search for Set origin to 3D cursor. In Numbers panel, clear (set to 0) cone's rotation as well as location, so it's inside the sphere, then with Ico.A selected, change its instancing to Faces:

And we got the cones:

Though there's still some room for additional cones - exactly where the Ico Sphere's vertices are... So let's hide the Ico.A and Cone.A, unhide the Ico.B, Edit the mesh, select everything, Ctrl+B for bevel, scroll down (mouse wheel) for minimum number of edges, C to clamp, and move your mouse to maximize the bevel like so:

M, B, increase the range until you remove all small faces:

Adjust Cone.B to the top face of Ico.B just was done with A variant. Don't resize the cone, though. Result:

I'd advise hiding both sphere, going into edit mode of each cone, hiding everything except the base face, then in object mode unhiding everything, selecting both cones, going to edit mode, and then scaling the bases so they have satisfying size:

Select Cone.A, Ctrl+A, M (Make instances real), Shift-click one of the cones to activate it, Ctrl+J to join those cones into a single mesh/object. H to hide it. Repeat for Cone.B. Remove the rest. Unhide the cones, select both, Ctrl+J to join them. Now you have the equivalent of the first screenshot in this post.
You can now create a "proper" sphere to your liking, e.g. UV sphere. You can add another sphere to cut the insides of the main sphere, you can add a cube to cut a fragment of the resulting shape... Once you're done, Hide the cones, as well as other objects you use for cutting (or change their display to bounds, as I did for the inner sphere and the cube) and add boolean modifiers like so (before doing so better save your project :D):

And here's an underperforming animation, notice the bugs...
