Yes, you’re correct, Reagan did use the term “socialized medicine”.
And for the sake of accuracy, in this 1961 speech, he opened by equating “socialized medicine” to socialism, and continued to the end with that theme:
Now back in 1927 an American socialist, Norman Thomas, six times candidate for president on the Socialist Party ticket, said the American people would never vote for socialism. But he said under the name of liberalism the American people would adopt every fragment of the socialist program....
{snip]
One of the traditional methods of imposing statism or socialism on a people has been by way of medicine.
[snip]
Now let us see what the socialist themselves have to say about it.
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From here it is a short step to all the rest of socialism.
And that is my point: "socialized medicine" equals socialism in health care. Therefore, since the term means socialism, quibbling over which word to use contributes nothing to the debate.