They do vary in size and the ones you had seem rather large to me. That could be a reason why the stems weren’t included.

Keep your eyes out for smaller ones. The size of caper buds doesn’t necessarily denote their quality, but the size of caper berries very definitely does. You only really notice the seeds in the bigger kinds, just like the ones you had. I think it’s because the less mature seeds are more of a tender hollow membrane, as they mature they become firmer and darker. I just looked back at your photos, you can see the difference in the second example between the smaller and larger caper berries you cut into.

My mum’s brother preserves the tender leaf shoots from capers, in Spanish they are called “tallos” (ta-yos), or rather, “tallos de alcaparra” I originally thought I didn’t like them (too bitter) but then I had some commercially produced ones which were milder in taste. Those I enjoyed. Even in Spain these aren’t so common and I’ve not seen them in London at all.