What is the difference between "a lead guitarist" and "a solo guitarist"?
Typically a lead guitarist plays melody lines with a rhythm guitarist and other instruments such as bass, drums, keyboard etc.
A solo guitarist, however, is always the only guitar, and is sometimes the only instrument (you can have a single person singing and playing solo guitar, or you can have guitar and drums etc.)
So the definition is simply to distinguish between the only one, or the lead one.
Here is my 2 cents worth.
The answer probably depends on the situation and context. It also probably depends on what whoever uses the respective terms mean.
"Lead Guitarist" is commonly understood to refer to the guitarist in a band who plays the "guitar solos" and fancier fills, runs, and licks along with a "rhythm guitarist" who maintains the rhythm by playing mostly chords.
Because a lead guitarist is the one who plays the guitar solo's - it's possible that some folks could refer to a lead guitarist as a solo guitarists. However in my opinion, that would be a misnomer.
The word solo means one. A part of a song that features a single instrument while the other instruments maintain the rhythm but stay in the background, is often referred to as an "instrumental solo".
But if I used the term "solo guitarist", I would be describing a solo musician who is playing without any other musicians (by themselves) and who's instrument was a guitar. I often play guitar to accompany my singing as a lone (solo) musician and I would refer to myself as a solo guitarist in that context. A classical guitarist who plays solo (by themselves) would also qualify as a "solo guitarist" by my definition.
I hang around and talk with many musicians and have over 100 musician friends and I am not aware of any who use the term solo guitarist to refer to the lead guitarist if there is more than one guitarist on stage at a time (even though the lead guitarist plays the guitar "solos"). However it is quite possible that in a band with only one guitarist - that one guitarist could be referred to as a "solo guitarist". But I think it would be less confusing in that situation just to refer to that musician as "the guitarist" or "the guitar player" - not the "solo guitarist".
You are free to refer to any guitarist any way you like. But if you say "lead guitarist", I am going to assume you mean the guitar player who plays "lead" or plays the parts of the song identified as "guitar solo". If you say "solo guitarist", I am going to assume there is only one guitarist and probably only one musician period (playing solo).
Unless of course you use that term in a context where there are clearly other musicians or other guitarist - in which case I will know that your definition of "solo guitarist" and mine - are not the same.
In terms of position or general role in a band I don't find much distinction between the term "lead" and "solo" guitarists. When was the last time you heard a guitarist who ONLY played solos?
One useful distinction between the two is compositional focus. If the part played by the lead guitarist becomes the focus of the song at that point, they becomes the "solo" guitarist for the duration of their part. Guitar players like Satriani and Paul Gilbert who have entire albums written around their leads can be considered solo guitarists for the whole album.
The other distinction comes from the accompaniment. If you're the "lead" guitarist, you lead the accompaniment. If you're a "solo" guitarist, you have no accompaniment and you play by yourself. Outside of this I find that the terms get used rather interchangeably.
There are some guitar players who have played unaccompanied. Some diverse examples: Joe Pass, John Fahey, Martin Taylor, Leo Kottke, Andres Segovia, Tommy Emmanuel,.... These players play what is broadly known as solo guitar music, and it makes sense to call such players solo guitarists.
In the larger musical world, players who play with bands or orchestras sometimes take over the focus of the music to play solos; these solos may even be the central feature of the music in some cases. These players are called soloists. Think Itzhak Perlman, Louis Armstrong, Sonny Rollins, Jacqueline du Pré, Jaco Pastorius,.... Guitar soloists include players like Jim Hall, Steve Vai, Steve Morse, Pat Martino, Joe Satriani, Wes Montgomery, Pat Metheny, Paul Gilbert, Grant Green,....
A lead instrument is one that takes over the melody at some point in a piece of music, and this is what a lead guitar does when it plays a solo. So, lead guitarist is just another term for guitar soloist.
"Lead Guitar" is a term used in 60s-style "groups" (like the Beatles) where Lead Guitar played the clever stuff, Rhythm Guitar strummed the chords.
A "Solo Guitarist" is a guitar who has a solo to play. Maybe completely alone, maybe in a concerto situation, maybe as a featured member of a guitar ensemble. Don't over-think it!