Reagan was for a social safety net. So were both Bush's. It's a fail to claim that Reagan was for socialized medicine. In fact, from your own link in Reagan's own words as he contemplates government assistance to help the poor access the private health system:
Regarding national health insurance, you could reassure your student, Miss Lee Catcher, that while I am opposed to socialized medicine, I have always felt that medical care should be available for those who cannot otherwise afford it. I have been looking into a program whereby government might pay the premiums for health insurance for those who cannot afford it and, at the same time, make such premiums for others a tax credit or deduction, preferably credit to encourage more use of private health insurance. There is also the problem of insurance for those catastrophic cases where the medical care goes on for years at a tremendously high cost. I proposed a form of government insurance for that in California when I was governor, but we couldn't get any legislative support for it. I do believe this is a particular problem which must be faced and where the government could have a hand.
I hope you have a very happy holiday season.