The second definition is a bit more comprehensive than the first, which only really deals with one kind of belief about “the rapture”. Pretty much all Christians believe in the event that some Christians call “the rapture”, but only certain types (pre or mid-tribulation rapturists) believe this:
“The rapture of the church is the event in which God removes all believers from the earth in order to make way for His righteous judgment to be poured out on the earth during the tribulation period.”
The texts which most clearly refer to the “rapture” event don’t actually describe that, it is an interpretation built from linking in other texts and applying them to this event. So, the “rapture” terminology can be confusing. Some are referring just to the event described in the Bible, and others are referring to that event, and a specific interpretation laid on top of it as well.
Thanks!