Checking for errors
Similarly to how we check for errors in the 250 box challenge, we're going to be extending our lines and analyzing how they converge towards their intended vanishing points. The only difficulty here is figuring out which lines to check for the cylinder.
The easiest one is just our minor axes - same as the previous checking method, find the true minor axis for each ellipse.
The other two are a bit more difficult. For each ellipse, it's got four points at which it makes contact with the plane that encloses it. This is assuming that you've got your ellipse perfectly snug within its confines, and not spilling out of it or falling short. If either of these last two options are the case, you'll have to best-guess where it would have touched the edge. There's still value in this.
These contact points can be grouped into pairs, as shown in this diagram. The members of a given pair will sit on opposite edges on the plane.
Draw a line that passes through both members of a given pair of contact points and extend it back into space, towards its vanishing point. The VP will be the same as one of the box's other two remaining vanishing points, so you can check it directly against its lines.
What makes this so much trickier than the extension method for the boxes alone is that there's a lot that could be going wrong. First off, your minor axis alignment could be off. Secondly, you may not have drawn the ellipse to fit snugly within the box. Thirdly, the planes you drew your ellipses in may not really be squares in 3D space.
This error checking method serves to help us work on gradually building a more intuitive sense for the proportion of these forms. Don't get stressed if you're constantly turning up mistakes - it's entirely normal, especially with so many different factors to control. The point is to gradually get better.
The truth of the matter is that for the most part, for any given drawing, you only need to be close enough. These correction methods go much farther than that, making every single error excruciatingly obvious. This in turn helps us learn from them a lot better, but don't think that you're going to be held to this standard in your actual drawings.