Generally speaking, a one month period extends from a day in one month to the same day in another month. Although it's possible a particular agency will interpret it differently, it's likely not the case. (But the only way to be sure is to ask them.)
However, in interpreting these instructions, you can remove any part that you know not to apply.
Since the card will expire on June 20, 2019, it has not yet expired. Therefore:
If you are a permanent resident whose 10-year green card
has expired or will expire within the next 6 months, you may begin the renewal process.
Assuming that the interpretation of within the next 6 months is correct, then you could have begun the renewal process on December 20, 2018, which would have been the start of the 6-month period. You can also begin the renewal process at any time now.
There is also this:
Additionally, if you applied for naturalization at least six months before your Green Card expired.
It's not clear if this only covers past events. But if it also cover future events, then it could be rephrased in this way:
You may apply for naturalization so long as your Green Card will not expire in the next six months.
June 20 is less than six months from now—so it will expire in the next six months.
If you had wished to apply for naturalization, you must have done so before December 20, 2018.
(Although, now that I read it again, it's not clear if this is something separate or an additional requirement of the renewal process.)
But, again, this is just one interpretation of the instructions. To be perfectly clear, the agency should be asked—not random people on the internet.