Let me make an attempt in answering a so-commonly-asked question the best I can.
Say we have an Arduino - A and we want to digitalRead a signal from a pin A2.
We will later reference Arduino 5v pin as A(5v) and its ground pin as A(gnd)
Let's me make a simple connection below and read the digital INPUT on Arduino Serial console:
Connection 1: A2 to A(5v) will read 1
Connection 2: A2 to A(gnd) will read 0
Fact, if we were to add a button B in between to the above simple connections in a pressed-down state nothing will change to the INPUT digits.
Connection 1: A2 to B(pressed) to A(5v) will read 1
Connection 2: A2 to B(pressed) to A(gnd) will read 0
But what will happen when the button is released?
Will the digitalRead always guarantee to deliver the opposite digit? No! But do we want to? Yes, of course!
To ensure the reversed digit is delivered when the button is released we would need to add a Resistor - R to the above respective connections as follows:
Connection 1: A2 to R to A(gnd) will give an opposite read 0 when button NOT pressed
Connection 2: A2 to R to A(5v) will give an opposite read 1 when button NOT pressed
Now you can finally see the answers for yourself.
a. Connection 1 is PULLDOWN coz Resistor pulls it DOWN to the ground to give you 0 in OPEN state.
b. Connection 2 is PULLUP coz Resistor pulls it UP to the 5v to give you 1 in OPEN state.
Oh wait, now you know why in the PULL-UP the button delivers 1 when open. :)