The first is correct. But the terminology "carrier signal" and "modulating signal" can be confusing because of the word "signal". I prefer carrier "wave" and signal "wave". The signal "wave" is the "message" thats being sent, the "carrier wave" carries that message.
There are 2 types of modulation:
1. Frequency modulation FM
http://www.m2m.ecs.soton.ac.uk/Wcc5e5a23ddd1e.htmFor FM modulation, the frequency of the carrier wave is varied by the signal wave
Here the amplitude of the carrier wave is NOT changed by the signal wave. Draw the signal wave over the top of the carrier wave. Basically the frequency of the carrier wave inside the "negative" part of the signal wave is increased, and decreased in the "positive" part of the signal wave.
2. Amplitude modulation AM
http://www.m2m.ecs.soton.ac.uk/Wc63e59ed037ae.htmFor AM: draw two modulating sine waves, one the with the negative amplitude of the other. These two waves form the "envelope". The carrier wave fits inside the envelope ie the regions bounded by the two signal sine waves, and keeps its frequency. Therefore its maximums will be the maximums of the modulating waves, its minimum will be the intersections of the two modulating waves.