If we trust Dumbledore, Riddle had already killed his father long before he made the ring into a Horcrux.
Dumbledore deduces that the ring was turned into a Horcrux only after Voldemort had decided that he no longer wanted to wear it. Since he was wearing it when he asked Slughorn about Horcruxes then he must have killed his father and captured the ring before he made the ring into a Horcrux. He then subsequently returned to the Gaunt shack and deposited his new Horcrux there.
"I stumbled across the ring hidden in the ruin of the Gaunts’ house. It seems that once Voldemort had succeeded in sealing a piece of his soul inside it, he did not want to wear it anymore. He hid it, protected by many powerful enchantments, in the shack where his ancestors had once lived (Morfin having been carted off to Azkaban, of course), never guessing that I might one day take the trouble to visit the ruin, or that I might be keeping an eye open for traces of magical concealment."
(Half-Blood Prince, Chapter 23, Horcruxes).
Of course, Dumbledore isn't 100% right 100% of the time but he states himself that his educated guesses turn out to be true more often than not, and out-of-universe he's used as an undisputed source of new information, so I think we can trust his deductions here.
Remember that Voldemort's primary motivation in visiting the Gaunt shack and killing his father wasn't to find objects to make into Horcruxes. He didn't know about the ring or the locket until he visited the Gaunt shack. Indeed, he didn't even know that the Riddles lived nearby! He got all of that information from his conversation with Morfin, which is what led him to kill his family and steal the ring. But he only went to the shack to try and track down his living wizarding relatives. Everything else he did stemmed from what happened when he got there, but finding Horcruxes wasn't his motivation.
It may be helpful to map out the precise order of events.
At an unspecified time, probably quite soon after discovering that he was a wizard, Voldemort commits to finding and killing his father as revenge for his upbringing.
“He left her and returned to his Muggle parents before I was even born, Potter, and she died giving birth to me, leaving me to be raised in a Muggle orphanage...but I vowed to find him...I revenged myself upon him, that fool who gave me his name...Tom Riddle.”
(Goblet of Fire, Chapter 33, The Death Eaters).
Later on, Voldemort begins looking into his family history, focusing on his father's side.
“Those whom I could persuade to talk told me that Riddle was obsessed with his parentage. This is understandable, of course; he had grown up in an orphanage and naturally wished to know how he came to be there. It seems that he searched in vain for some trace of Tom Riddle senior on the shields in the trophy room, on the lists of prefects in the old school records, even in the books of Wizarding history. Finally he was forced to accept that his father had never set foot in Hogwarts."
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
After those searches come to nothing, he looks into his maternal ancestry.
"I believe that it was then that he dropped the name forever, assumed the identity of Lord Voldemort, and began his investigations into his previously despised mother’s family."
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
This leads him to discover the Gaunts.
“All he had to go upon was the single name ‘Marvolo,’ which he knew from those who ran the orphanage had been his mother’s father’s name. Finally, after painstaking research through old books of Wizarding families, he discovered the existence of Slytherin’s surviving line. In the summer of his sixteenth year, he left the orphanage to which he returned annually and set off to find his Gaunt relatives"
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
As the quote demonstrates, this was in the summer of Voldemort's sixteenth year.
Voldemort goes to the Gaunt shack and introduces himself to Morfin - the sole surviving Gaunt - as a wizard (through the medium of Parseltongue).
Voldemort’s eyes moved slowly around the hovel and then found the man in the armchair. For a few seconds they looked at each other, then the man staggered upright, the many empty bottles at his feet clattering and tinkling across the floor.
“YOU!” he bellowed. “YOU!”
And he hurtled drunkenly at Riddle, wand and knife held aloft.
“Stop.” [in Parseltongue]
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Morfin, inadvertently, tells Voldemort about the family's ancient artefacts - the ring and the locket.
Morfin pushed the hair out of his dirty face, the better to see Riddle, and Harry saw that he wore Marvolo’s black-stoned ring on his right hand.
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
“Ar, he left her, and serve her right, marrying filth!” said Morfin,
spitting on the floor again. “Robbed us, mind, before she ran off!
Where’s the locket, eh, where’s Slytherin’s locket?”
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Morfin also tells Voldemort that his relatives live nearby, and which house they live in.
“I thought you was that Muggle,” whispered Morfin. “You look mighty like that Muggle.”
“What Muggle?” said Riddle sharply.
“That Muggle what my sister took a fancy to, that Muggle what lives
in the big house over the way,” said Morfin, and he spat unexpectedly
upon the floor between them.
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Voldemort ambushes Morfin and goes up to the Riddle house to murder his family.
“When he awoke next morning, [Morfin] was lying on the floor, quite alone. Marvolo’s ring had gone.
“Meanwhile, in the village of Little Hangleton, a maid was running along the High Street, screaming that there were three bodies lying in the drawing room of the big house: Tom Riddle Senior and his mother and father."
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
After killing his father Voldemort returns to the Gaunt shack, steals the ring and frames Morfin for the murders.
"Then he returned to the Gaunt hovel, performed the complex bit of magic that would implant a false memory in his uncle’s mind, laid Morfin’s wand beside its unconscious owner, pocketed the ancient ring he wore, and departed.”
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Note that Voldemort departs the Gaunt shack after killing his father.
Voldemort begins researching Horcruxes at Hogwarts, probably using the Hogwarts library as his main source of information.
"This is the one that gives explicit instructions on how to make a Horcrux. Secrets of the Darkest Art - it's a horrible book, really awful, full of evil magic. I wonder when Dumbledore removed it from the library... If he didn't do it until he became headmaster, I bet Voldemort got all the instruction he needed from here."
(Deathly Hallows, Chapter 6, The Ghoul in Pyjamas).
Voldemort may have actually started his research into Horcruxes before he ever encountered the Gaunts. There isn't any definitive information which tells us whether or not his genealogy research predated his Horcrux research, as far as I'm aware. (Hat-tip to DariM for the quote.)
Voldemort asks Professor Slughorn about Horcruxes. He's wearing the ring, which (as the question correctly points out) means that he's already killed his father.
His right hand lay negligently upon the arm of his chair; with a jolt, Harry saw that he was wearing Marvolo’s gold-and-black ring; he had already killed his father.
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Voldemort turns the ring into a Horcrux.
"I stumbled across the ring hidden in the ruin of the Gaunts’ house. It seems
that once Voldemort had succeeded in sealing a piece of his soul inside it, he did not want to wear it anymore."
(Half-Blood Prince, Chapter 23, Horcruxes).
Voldemort returns to the Gaunt house and hides the ring there.
He hid it, protected by many powerful enchantments, in the shack where his ancestors had once lived (Morfin having been carted off to Azkaban, of course)...
(Half-Blood Prince, Chapter 23, Horcruxes).
The key thing to point out from this order of events is that there were actually two trips which Voldemort made to the Gaunt house. The first was simply a family research trip. When the opportunity for murdering his father and recovering an ancient family artefact occurred he took it with both hands. Crucially, after getting what he came for, and having killed his family and acquired the ring as a bonus, he left the Gaunt shack.
"Then he returned to the Gaunt hovel, performed the complex bit of magic that would implant a false memory in his uncle’s mind, laid Morfin’s wand beside its unconscious owner, pocketed the ancient ring he wore, and departed.”
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
Creating the ring-Horcrux came later, as did depositing the ring in the Gaunt shack. This required a second trip. Needless to say, in between the two trips Voldemort had found out all he needed to know about Horcruxes and successfully created one of his first Horcruxes. It's not known how much of a gap there was between his first and second visit but it would need to be enough time for him to gain the sufficiently advanced technical knowledge required to make a Horcrux, and to exercise his significant talent by actually successfully performing the relevant spell. That would have taken a while.
Nevertheless, to address the core of the confusion in the question: no, Voldemort hadn't had that conversation with Slughorn when he first acquired the ring and murdered his father. That was his first visit, and he placed the Horcrux in the Gaunt house on his second.
