The whole "přítel/-kyně" word is a little old-fashioned today. When it's used, it almost always means "boy/girlfriend", but even in that sense it's not used much in colloquial language.
The most common word for "friend" is "kamarád/kamarádka" or (more slang) "kámoš/kámoška".
And there are many ways to say "boyfriend/girlfriend" in varying degrees of slang, many of them must be accompanied by a possessive pronoun: "můj kluk / moje holka", "můj milý / moje milá", "můj starej/mladej / moje stará/mladá" (the last ones are more often used for husband/wife or at least when the relationship is very serious).
Having said that, the word "přítel" belongs to basic Czech vocabulary. It has a serious tone about it in today's Czech - when it's not used as "boy/girlfriend" then it means "trusted/sincere friend", not "buddy" or "acquaintance".
In official communication the word přítel/přítelkyně is still the way to go. I heard it several times on TV in natural communication in the past days (and I do not watch TV too much). There are situations where kamarád/kamarádka is simply too colloquial or may indicuate a casual friendship. For a lifelong friend of people of certain age (and social status) the words přítel/přítelkyně are more appropriate.
Take this example. Eugen Ionesco, Scéna ve čtyřech - a theatre play from 1959, translation from French to Czech by Pavel Trtílek in 2006.
Všichni tři muži přestanou a otočí se k Hezké dámě.
HEZKÁ DÁMA: Pročpak se hádáte? (roztomile) Drazí přátelé…
DUPONT: Drahá přítelkyně, konečně jste tady, vy nás jistě dostanete z této šlamastyky.
DURAND: Drahá přítelkyně, uvidíte, kam až nás dostala nebetyčná podlost.
MARTIN (přeruší Duranda): Drahá přítelkyně, pojďte, já vám to všechno vysvětlím.
DUPONT (Martinovi a Durandovi): To já jí všechno vysvětlím, protože tato půvabná dáma je moje snoubenka…
Have you read the previous discussion?
It is more typical for older generations or even maybe for the first half of the 20th century and before.
Using ty or vy is the T-V distinction. It shows the level of familiarity, not closeness or affection. In the 19th century people still used V for their parents. And often for their lovers.
I did read the discussion. Ty has always been described to me (by numerous Czech people) as the form you use with your friends whereas vy is used to address an elder or someone you just met. Because Zofie is a friend, I assumed ty to be the most fitting and wondered if I was missing something.
Ty is used with people you are on the T side of the T-V distinction. This could be your relatives, friends or enemies. One can be a good friend of someone and still keep the V side. That used to be more common that it is now.
BTW, using the first name and V at the same time is quite common.