Is the new spindle a deep spindle too? Does not sound like it, but wanted to be sure.
Since your old one had the larger seal (which I thought was the early design, but apparently is the later design) then that is the more desirable one because your axle is sized to go into the large seal.
If the new spindle is the deep type that matches yours, you would want to use the larger wheel bearing type seal and press it in to the back of the spindle like your original. If that means sourcing one locally to get you going, that's what I'd do. If you have time however, I would contact the vendor (was that us by any chance?) and see if they'll send you the correct seal.
But if the new spindle is the shallow type then you have to use the included older style seals with it. Hopefully it works, and I think it should, but have never actually tried it myself.
Did you also get the plastic thrust washer? Has a bevel on one inner edge and is usually plastic even though the originals were bronze and fits over the axle shaft?
The way these are laid out is the bearing is recessed into it's pocket, then the small seal is a loose fit just on top of the bearing. I've always installed them with the open face inward toward the bearing. But many here do it the other way and I don't think I've ever heard of an issue. Not that we'd usually know of it, but after all these years I would think any problems would have cropped up already.
Then the thrust washer slides over the axle first, bevel facing the differential.
The larger seal shaped kind of like a bellows will also go on either way, but I usually install them with the cupped side facing the spindle and the more solid side facing the axle flange and thrust washer.
Do a trial fit to see how this works out. If it just does not seem like it's going to work, you may need spindles with the deep well and the big seal. Which would be a bummer and more of a hassle. So hopefully the new layout will work.
Paul