So when Tobymac specifically speaks to God in "Steal My Show", asking Him to take control of his work, a concept that is consistent with Christian teaching (and excluded by many other religions), by saying:
"My Life, my friends, my heart, It's all yours, God, take it away, my dreams
my fears, my family, my career, take it away, It's all yours, God
Take it away, take it away, It's You I want to live for"
and you know he professes to be a Christian, you assume it's general theism? I do not understand that.
It becomes particularly hard to understand when I look at the lyrics of some of the other songs you posted.
One Thing Remains never even mentions God, much less Christ. If you make a negative assumption about Toby, why do you make a positive one here, on less direct evidence? Oh, certainly you can grab the Christian context of the words "grave" and "debt is paid", but without clearly saying God or Christ, isn't this just another song that could be sung to a person, or another deity?
A City On a Hill never mentions Christ, only using Lord. From your logic, it could just as easily be about the Muslim God. Nothing Christian distinctive about that song.
I fear you are falling into the trap of letting musical style and preference influence how Christian you feel a song to be. I also see that issue in our church, and ironically in another thread you're participating in. The one if the poster who called anything not liturgy music to be "secular". You seem to understand the problem there - it's just a different point on the same path.