President Carter finally made an old-fashioned partisan speech to the late mayor Richard Daley's Cook County Democratic organization, but he scrupulously protected his own downstate political interests in a flashy display of ambiguous Carter-style politics here last week.

The ambiguity was both typical and intentional. With his huge strength in Chicago's teeming black wards a political constant ("Jimmy has a mystical link with blacks," one white legislator from Cook County told us), the president had little need to woo the once impregnable Daley organization.

Now weakened and facing more troubles in the future as Daley's former lieutenants fight for power, the country's last political machine is perceived by Jimmy Carter's political aides as not quite superfluous but surely as of declining value.

Carter is a different political breed from such predecessor national leaders as Sen. George McGovern, the late Hubert Humphrey and former president John F. Kennedy. He is not personanon grata downstate, where antipathy for the Chicago Democratic machine is expressed in language unfit for the family dinner table.

Having passed up last year's annual fund-raising dinner here hosted by Cook COunty Chairman George Dunne, Daley's successor as party chairman, Carter allowed Dunne's invitation to this year's May 25 dinner to languish unanswered on his desk for weeks. That infuriated the outspoken Dunne and led to juicy stories about the president's White House staff work, notoriously sloppy, and his failure to appreciate how dependent he is on Dunne's party organization.

In fact, however, the long delay in accepting Dunne's invitation was deliberate, an indication of the high importance given by the White House to the Baptist Belt in downstate Illinois, full of twice-born Christians far removed from Chicago's racial polyglot.

Reflecting the White House mood was this reply by a presidential aide when we asked him about fractured relations with Dunne. "The only trouble with George Dunne," he told us, "is George Dunne."

Indeed, it was not until he arrived here that the president finally dicided to give the frankly partisan speech wanted by Dunne for his $100-a-plate fund-raising dinner, packed with organization leaders (and a sprinkling of Republican businessmen long allied with Daley). Twice discarding speech proposals portraying himself in a more presidential image, Jimmy Carter at the last minute chose to dish out partisan red meat only slightly disguised by his bland delivery. For the first time in his administration he also departed from his anti-pomp-and-ceremony campaign to give specific approval for the playing of "Hail to the Chief" - a stirring symbol dear to the hearts of Cook County patriots.

But just 12 hours later, addressing a joint session of the Democratic-controlled legislature, Carter played a different tune. "I am a very conservative southern businessman by circumstance," he said in the course of a long, affirmative reply to a legislator's question as to whether he intended to push the labor-reform bill through Congress. Such a self-description would have been heresy the previous evening; downstate, it reminded Illinois farmers and small businessmen that Jimmy Carter, like them, is a small-town southerner. "He wants to remind them that he has a rural background, too, and isn't in with that Cook County crowd," one prominent Democrat told us.

Likewise, by unannounced prearrangement, he was asked a question about the Mideast arms deal by a Jewish legislator. In a long, careful reply, Carter defended a Mideast policy that is under emotional attack in Cook County but has enthusiastic approval in conservative downstate.

What makes such ambiguous politics possible is the president's unprecedented hold on Chicago's ever-rising black population. He counts on their voting for him in 1980 with or without Dunne's Cook County machine. Practicing the politics of diffusion may be risky, but Carter displayed a deft touch here. He departed the state more confident than when he arrived, convinced that the only way he could carry Illinois in 1980, after losing it in 1976, is to keep a certain distance from Dunne and the Democratic machine.