I've just experienced this very situation recently. Here's what I learned about U-style slots. I did not fully research T-slots, which is a whole other world of non-standard standards.
Imperial sized U-slots come roughly in two flavours: 3/4" and 5/8". The huge majority of after-market equipment out there, at least in the Americas, is one of those two sizes.
Of these two, 3/4" seems to be the most common nowadays by a huge margin, with older equipment (in my experience) being the smaller dimension. This makes it hard to match after-market equipment to my table saw (5/8"), but not my newer router table (3/4"), for example. So, if you pick up older equipment at auctions and yard sales, it'll probably expect 5/8ths tracks.
As far as I know, there is no standard, other than the ad hoc standard of these two sizes, with 3/4" by far more common.
A note on dimensions: U-slots are generally formed such they are not square. i.e., the cross-section dimension at the top will be smaller (by only a few thousandths) than the same dimension at near the bottom. This is to facilitate clamping/locking mechanisms so that lateral force tends to pull tracks down into the channel; that is, locking bars will wedge themselves in a bit. This is similar to how clamps and dogs can have surfaces offset a few degrees to keep work from squeezing out, or losing square as force is applied. U-slots also tend to have generous corner radiuses in the bottom to allow for free movement in the slot, and to allow room for locking mechanisms.
19mm is roughly 3/4 inch, and these slots are meant to be a bit sloppy, so perhaps the Euro sizes correspond to those, as well. 13mm is a little sloppy for 5/8ths, but it might be close enough. I mean, it depends if they are talking about the slot or the track, and if the track is meant to slide or hold during use.
As for how you know what ones you need? Well, in my case I need both. You may only need one of those sizes. Take some calipers or approximate with a ruler or scale and see what you have on your equipment. Like I said, these sizes are meant to be nominal, so there can be a lot of slop. And, I cannot stress this enough: it depends on whether the track is for locking or for sliding.
A note on U-track purpose: U-tracks have two purposes: you either want something to slide with a slight interference fit to act as a guide (e.g., a cross-cut sled, or mitre bar) or you want it to clamp and not move (e.g., finger-boards or other safety equipment). Sliding bars want a loose interference fit so things stay square. Locking bars can be a little looser because they have a locking mechanism (with a j-slot, or a cammed t-nut, or some other mechanism) that is intended to keep the bar from moving.
Milescraft after-market stuff that fits mitre slots often ship with both sized tracks. I was unable to source 5/8ths tracks anywhere but as a part of a kit from this company. They sell spare parts for their jigs and fences, so I am considering picking up more 5/8ths track from them.
You might just consider making wood tracks for some of your jigs and fences. I'm considering this for my own saw instead of buying them from that single source. This way I can keep the locking tracks on my fingerboards, and have a dedicated cross-cut sled that slides on hardwood tracks. Of course, this is yet another project one has to fit into the shop time they have...