As you can see, the AND operator drops every row in which at least one value equals -1. On the other hand, the OR operator requires both values to be equal to -1 to drop them.
That's right. Remember that you're writing the condition in terms of what you want to keep, not in terms of what you want to drop. For df1:
df1 = df[(df.a != -1) & (df.b != -1)]
You're saying "keep the rows in which df.a isn't -1 and df.b isn't -1", which is the same as dropping every row in which at least one value is -1.
For df2:
df2 = df[(df.a != -1) | (df.b != -1)]
You're saying "keep the rows in which either df.a or df.b is not -1", which is the same as dropping rows where both values are -1.
PS: chained access like df['a'][1] = -1 can get you into trouble. It's better to get into the habit of using .loc and .iloc.