Everything you are saying lines up with conception being between June 10-12. What about the situation is confusing you?
Perhaps you didn't know that the "weeks" count given to you by your doctor or ultrasound tech starts on the first day of your last period?
The GA (gestational age) count uses a 40-week count for the whole pregnancy time period, but a full-term pregnancy from conception to birth is only 38 weeks long. The medical world uses a 40-week count, to enable counting to begin on the first day of your last period. (Even though the first two weeks, you haven't conceived yet.) The first two weeks after the period are considered part of the pregnancy time period probably for historical reasons (back in grandma's day, ultrasounds were not available to tell how far along someone was, the only information they had was from her periods. So they would use the day the last period came as a starting point for the count). Despite ultrasounds now being widely available, this method of counting out the pregnancy time period still is how they do it. Any time you get a GA it will be a count that begins on the first day of the last period.
If you were to say to your doctor, "25 weeks? Does that mean I conceived 25 weeks ago?" he or she would say, "No, you conceived 23 weeks ago." A doctor knows you weren't pregnant in week 1, you were having a period! But the count is still done that way and everything medical is calibrated to it, which obviously can be very confusing for women who are trying to decide who is the dad.
My advice is, only count back from your estimated due date from your ultrasound(s) when trying to calculate a conception date, don't get wrapped around the wheel trying to assess the "weeks" count for that purpose. The ultrasound saw and measured the actual baby, and that is the most accurate information you have. From your estimated due date, count back 266 days, and that is your estimated conception date.