To me, both "i am running out of strenght" are mixed voice. The first one has a bit more meat but the difference is subtle. What i refered as "pure chest" would indeed be a very overdrivy sound on opened wovels (on the audio, you are using covered wovels, which automatically throws you in a certain amount of mix).
Yes, the overdrive must feel like very effortful at the beginning, and any vocal tract shifting (being too harsh on consonants) will make the voice yoddle in a bad way. But with the propper support and phrasing, you will be able to go up there. That may not be the sound you are searching for, but as an exercice it will help you getting a heavier mix.
This stuff is quite Advanced and it must be monitored and never be so harsh that you can feel it the next day.
You already got a good light mix and good head resonance. I feel what you just need now is to put just a bit more compression to the sound.