I forget how the form ask the question but you should put down October, if it ask what month you want benefits to start. I re-wrote this using your mid-October date from what it said in the link. ⬇️
If you reach full retirement age on Oct. 15 and start benefits then, you’ll get your first Social Security payment on the third Wednesday of November.
excerpts from the above link:
your first payment generally will arrive in the month after you attain your full retirement age. That’s because Social Security pays benefits a month behind, so your benefit for, say, December 2020 will arrive in January 2021.
If your birthday is between the 11th and the 20th, the payment comes on the third Wednesday of the month.
So, if you reach full retirement age on Nov. 15 and start benefits then, you’ll get your first Social Security payment on the third Wednesday of December. ⬅️
There's an exception if you were born on the first day of the month.