If the signal is very faint, does the receiver still produce such a good signal ?
Yes it seems that the signal is very good even though I removed the external antenna from the transmitter (and receiver to see how it behaves). But at the moment the receiver and transmitter are very close to each other and I cannot move those further away of each other so that might be the reason that the signal looks good even though there is no antenna.
Those receivers have an auto-gain. If the transmitter is not transmitting, then the gain in the receiver is increased until noise is received
Thanks for this piece of information, now I think that I understand what is happening in the beginning of the receiving. When the transmitter is not sending anything the gain of the receiver is high thus it is receiving noise, but once the sender starts to send data the receiver decreases gain automatically because it can receive the data with lower gain and that is the reason that in the beginning the receiver is receiving also other data than the 1kHz preamble (?) that the sender is sending. This is visible in the attached Preamble.png i.e. that receiver is receiving first plain noise (with high gain), then some noise spikes in addition to the actual signal (lower gain) and eventually the receiver is receiving only the data that the transmitter is sending (minimum gain).
What the RadioHead library does, is trying to make sense out of the noise
It is doing it very well, at least I am able to send “Hello, world!” between two Arduinos without any issues