meaning is usage
And nonsense is popular, and platitudes aren't very helpful. Otherwise 'Scrotter fur thysunf, yous attenlow denge bitronic foum' should be very meaningful since I used it. I know, I know...
The point is that their characterisation of God is based on rational, philosophical argument rather than scripture or revelation. That's what natural theology is.
No, what you're calling natural theology. What you describe most academic philosophers would call deism. Consider
Natural theology, once also termed physico-theology, is a type of theology that provides arguments for the existence of God based on reason and ordinary experience of nature.
The latter is very important, consider the lilies of the field as evidence kind of stuff. It seems like you would agree with Voltaire.
"If God did not exist, he would have to be invented." But all nature cries aloud that he does exist: that there is a supreme intelligence, an immense power, an admirable order, and everything teaches us our own dependence on it.
It was an attempt at an empirical basis or focus for theology, until natural philosophy took prominence and went in entirely different directions.
Are you denying that 'natural theology' is a thing?
Well, virtually every philosopher since Kant will have serious problems with it, though really as early as Xenophanes depending on your preferences. I would say 'natural theology' is a fossil phrase, about as current today as natural philosophy, not quite as antique as Quadrivium though.
I'm not saying that there's one set of doctrines endorsed by everyone who goes under the label 'natural theology'.
this seems to contradict
'The God of the Philosophers' is a phrase which usually refers to the being which the medieval philosophers... identify as God
When they argue and contradict each other. Perhaps a silly point.
OP seems to be asking if most philosophers in natural theology (note the word 'usually') had arguments for the omnibenevolence of God. In short, they're asking for rational arguments as to God's omnibenevolence, rather than arguments based on scripture or whatever.
Fair enough, though I believe most of them come down to a belief in 'god is good' more or less directly from scripture. Or genesis, god made such and such and saw that it was good (paraphrasing obviously). Maybe it's out of John in the NT, I don't recall.
Most of the 'arguments' for the benevolence of god (and gods) I believe usually fall under providence. Describing the bounty of nature as divine providence, maybe relying on arguments for design. In the genesis tradition god made creation for man. That's all I have off the top of my head at the moment. It may fall under divine revelation from scripture for some of them, especially those who didn't feel that it required buttressing by reason.
I think few have attemped any direct rational arguments for divine benevolence. Leibnitz made some sort of arguments for it in his 'Theodicy' (I hope you like Monads), but Voltaire lampooned it pretty mercilessly in Candide.