From a modern perspective, the type of pillow you chose (especially if both choices are roller pillows) is mostly based on your personal preference and working style. Traditionally different styles of pillows were used in different regions, but there isn't one best or worst type of pillow. There are also modern flat pillows with moving parts that act like a roller pillow by sliding the center part up.
The pillow must:
- Be wide enough for your lace pattern.
- Be stiff enough that any pin you put in stays where you put it.
- In case of tape lace the lace pattern must be long enough to fit once around the pillow (it can be longer, but cannot be shorter).
The pillow should:
- Rest at a height that is comfortable for you to work in.
- Be stable enough that it doesn't roll when you don't want it to.
- Ideally be big enough (in width and circumference) that all bobbins can rest on it in an orderly way.
The pillow in the first picture has the advantage that you can rest your arms on it while you work and that the bobbins lay vertically. If you work with a whole lot of bobbins, they can rest on the sides as well as the bottom. With the pillow style shown in your second picture the bobbins hang down vertically. When using that with a lot of bobbins, they bunch up and in the worst case they tangle.
Compare these two:
image source
image source
But these are extreme cases. While working with only a few bobbins, there shouldn't be much of a difference.
I think modern flat sliding pillows are the most versatile for a beginner because you can do flat works like doilies or lace collars on them as well as lace tapes. And they are smaller than most alternatives to stow away or take with you on a trip.
Both styles of pillows in your pictures are traditional forms (in different regions) and have been used for centuries, so you can expect them to have worked well for centuries. The big rolls may also be easier to DIY (again: people have been doing this for centuries).
I'd say in most cases the width of the lace is the deciding factor when choosing one of the traditional pillow styles. For narrow lace tapes or small doilies both work fine. But for wider patterns (like a wide collar, stole or curtain) you need a wider roll. At a certain size it just gets more practical to work with one big roll instead of a smaller roll inserted into a wider pillow.