Now..
It could be said that the word 'God' with capitalization..
Represents the God of the Bible..
As it does say in the Bible..
In Exodus chapter 20:1..
And God spoke all theses words, saying:
I am Jehovah your God, who has brought you out of the land of Egypt.
You shall not have any other gods beside Me..
Big G little g...
As according to the Bible...
No "other gods;" they are called such not because they are so, either by nature or by office (Ps. 82:6),
but because they corrupt hearts of men make and esteem them such—as in "whose god is their belly" (Phil. 3:19).
"Before Me" or "My face," the force of which is best ascertained by His word to Abraham,
"Walk before Me and be thou perfect" or "upright" (Gen. 17:1)

So.. If the Pledge of Allegiance...
Said one nation under Jehovah God or money said in Jehovah God we trust.
Instead of in God we trust or one nation under God..
Some definition of a name of which God do we trust in....
But that is not defined on a dollar bill or in the pledge of alligence..

So I guess it would be more assumed that thats what the
word God meant..
But yes.. They did not include any specfic name of a God..
Just the title of one instead...
Yet with a big G...