You have to be careful to interpret your statements 1 and 2 correctly.
In statement 1 you are talking about an infinitely long wave train which is made up of only one single frequency.
As soon as you have a wave train of finite length then the wave train is made up of the sum many frequencies.
So in statement 2 what you think of as a pulse of a wave with a frequency 2 Hz is actually a pulse of the superposition of very, very many frequencies which all add up to look like a pulse of a wave with frequency 2 Hz.
This subtle difference does not matter as long as the medium is no dispersive, ie the speed of the wave does not depend on its frequency.
If the medium is dispersive then the different frequency components that make up your 2 Hz pulse travel at different speeds and the shape of you pulse changes. As the shape of you pulse changes which part of it do you use to measure its speed?
I think that conceptually this is very difficult.
There are many animations on the Internet which try and show visually what happens.
Here are a couple:
Link 2 which gives you more control.
There are many others and I would be glad to hear of any ones which are better.