You can also use the Reset menu under the Memory menu to reset the calculator back to default settings without deleting anything.
The TI-84 Plus has no concept of background programs. (It does have a concept of hooks, which are sort of like
DOS-style TSRs, but that's not what's happening here.) To add some more information, the graphing screen is considered a separate "app" with its own configuration. BASIC programs can command the graphing screen to do things that change its configuration, and the graphing screen retains that configuration until the user or another program changes it. This is intentional design, since it lets users share data between built-in functions and user-created programs.
This model of sharing data between apps and letting BASIC programs reconfigure your calculator is a holdover from the late 70s to early 80s 8-bit
BASIC microcomputer vibe the TI-84 Plus has. It's actually reasonably powerful and convenient once you're proficient (as long as you like 80s microcomputers, anyway). Back then, computers didn't have the CPU power and memory for more modern concepts like running more than one thing at a time and having a shared clipboard. Early popular home computers didn't even have persistent storage; if you turned the computer off, you had to type in the program you wanted to run again!
The TI-84 Plus shares quite a lot in common with TI's first graphing calculator, the TI-81, released in 1990. It should be unsurprising that the TI-81 gets a lot from the era of BASIC computers; one suspects many of the engineers that worked on it first learned to program on an 8-bit computer. Up until the 80s, it was widely believed by engineers that all computer users would have some degree of familiarity with computer programming, and the idea of users buying commercially-developed software was somewhat a novel concept when Microsoft was founded. A lot of the software design of the TI-84 Plus still reflects an ideal that all students should learn some computer programming, and thus nearly anything you can do manually on a TI-84 Plus can also be done by a BASIC program.
Though I'm not even 30 years old, I'm sure the idea of math classes incorporating computer programming made perfect sense in the 90s, and I still think it's a lost opportunity. Sadly, it doesn't seem to have caught on. Ironically, it's probably at least partly TI's fault for keeping the TI-84 Plus's price way too high, as they could have profitably sold it for $50 in 2004, an era when even a single shared family computer (much less Internet) wasn't something public school teachers could count on students having.