I believe Jepps expire at 9:00 Zulu on the date indicated. This is probably the same no matter who the database/charts are from. If so, the dates in the FMS would be correct. The first database expires on the 4th at 9:00Z and the second one starts at 9, so both would include the date of the 4tg because both are active at some time during the 4th.

Note that this post is based on assumptions and bears no resemblance to real world activities. Personally, I would probably just choose the newer database if they are both active on that day.