Chest voice is not a “sound”, it is a register of the voice. The term came from singers noticing a sympathetic vibration taking place in the chest on lower pitches. There is a term called
full voice, though. It means that the vocal folds are adducting properly and that it is a full tone, not breathy, not raspy, not harsh, and not yelling or screaming.
For many years, there has been a thing between chest voice and head voice but untrained singers cannot always use it or find it. It is called middle voice. Some people call it mix or mixed voice or blend or other terms. I have used the term middle voice in training singers but I think it is good to only temporarily think about registers in the voice, especially when performing. When your voice is properly trained, you can connect, so to speak, the registers: chest, middle, head, and flageolet (as classical singer might say).
Whistle tones or superhead voice are upper register tones above head voice and can be heard as done by Mariah Carey, Rachelle Ferrell, but not many others.
If you only have chest voice and falsetto, you are missing a usable register in between. It has also been called the passaggio . The term has been mis-defined as “passage area”. It is an Italian word, as are most musical terms, and it simply means passage or transition. Singing in it can feel precarious and uncontrollable, if not impossible to use unless and until you are properly trained to use it. A very few people naturally can use it. In my experience, those who naturally or very quickly use it have been Italian or descendants of Italians. Why they are gifted this way, I have no idea but I do have a few theories.
If you ascend in pitch, chromatically from chest voice, and you suddenly flip or crack or break or disconnect in tone quality, there is an issue causing that and it takes training to overcome the habits which cause that issue to be perpetuated. Inside your larynx, if we were to examine the vocal folds with a scope, we would see that adduction is lost in the passaggio, if you are not able to maintain your sound as you go through the range of middle voice.
Most people have the problem of the larynx rising (some severely and/or suddenly so) and then try to force the notes out by screaming. You have discovered and would agree that screaming is not the answer for you. There are several vocal exercises which can help with your situation.