Thanks for posting that, it makes it a lot easier to figure this out.
Some of the videos you have in Good and in Bad are using the same basic model of lyre, so it's really less a given model you don't link than that it's someone's playing or recording style. Also note too that when you first get your lyre, you can improve any twanging or buzzing sound by asking for advice here on how to adjust it.
In my opinion, you can go either of two ways here:
Option 1: get a basic 16 like in most of your clips
Go for one of the larger steel-strung import lyres in the 16-string size (several of the clips you have include those). There are basically three different models of 16-string you commonly run across: the standard teardrop, the solid teardrop, and the "ears" model.
While being sold under different brand names, these are generally identical products from the same factories, so you don't need to buy these exact models/links, just any item on Amazon resembling them. I would try to buy from a seller with good feedback on that item, and one that offers Free Returns so you can send it right back if you don't like it or get a defective one. Though I would suggest you ask the sub first if a given problem is a temporary one while the new lyre is settling in, or something more serious.
Option 2: Go nylon
The other way to go, is to either get a nice nylon-strung lyre right away, or get a cheap model and set it up to be similar to the nice one while you save up money to potentially get a nice one. Check out these examples and tell me if this is more what you like or less what you like:
If you like the sound of these 10-string nylon-strung harps, you can either save up to get that same model (Marini Davidic Harp, $380 new, made in Pennsylvania), or get an inexpensive import 10-string and convert it to nylon strings, which isn't terribly difficult, just $15-20 for a set of Kinnor strings and an hour of messing with strings.
For a 10-string (steel, can convert to nylon), the Vangoa is the best-seller on Amazon and is $59, or you can shop around for any other 10-string that has lots of good reviews. Just DON'T buy a 10-string made in Pakistan, which you can identify by the terms "rosewood" or "lilacwood", crude deep carvings, gratuitous and inaccurate references to Ireland/Scotland, etc. Like don't buy one that looks anything like this or this and you'll be fine.
So those are my main two suggested courses.