If the initial reaction of some city officials to President Carter's plan for increased home rule sounded a little lukewarm last week, it's because in the beginning, most city officials - through no fault of their own - really didn't know what they talking about.

White House aides called it a simple and honest omission. The news summary and fact sheet announcing Carter's proposals did not include the President's endorsement of "removal of the federal government from the District's budgetary process by 1982," and expansion of self-government long-sought by home-rule advocates.

It was only after Vice President Mondale, who announced the plans, left the press briefing that a reporter asked Presidential Assistant Martha M. Mitchell about budget autonomy and the omission was discovered.

"What?" exclaimed Del. Walter E. Fauntroy (D-D.C.), when a reporter asked him for a comment. Fauntroy had read the same incomplete press release given reporters. He had already prepared and issued a statement saying he was hopeful that sooner or later Congress and the White House would get around to urging "complete self-government for the District, including full budget authority." Fauntroy, newly enlightened, amended his original statement in which he said he was appreciative to Carter, to say he was now "even more appreciative."

Mayor Walter E. Washington had discussed the proposal the previous evening in a telephone conversation with Mondale. But among those who prepared the mayor's statement, it was uncertain just how much budget autonomy the White House was going to urge. Thus, the mayor, in his response asked that one of the specific priorities, in Carter's announced general push towards "reducing federal intrusions" in local matters "should be a move toward greater budget autonomy."

City Council Chairman Sterling Tucker found out about the budget autonomy provision from an aide who found out about it from one reporter who learned about it from another reporter. Tucker called that provision the "sleeper" in the Carter package.

From the point of view of the city's elected officials, those 12 omitted words can easily be condensed into two: budget autonomy. Despite the home rule charter, it is actually Congress that balances the D.C. budget because it is Congress that has the final line by line say-so on what the city and cannot do with its money.

Thus, when city leaders, feeling Washington is already over-policed, for example, reduce the size of the police force, Congress beefs it back up. City officials overwhelmingly endorse plans for a downtown convention center and two members of Congress - neither of whom has a single District of Columbia citizren among his constitutents - overrule the city's leadership.

It is a situation that belittles the power of the city's elected officials in no uncertain terms and one that most city officials despise. So the Carter administration's plans to end it - even though they were announced through the back door - were nevertheless welcomed in city hall.

To be sure, the city probably will not get complete budget autonomy. James Dyke, an aide to Mondale and one of the two White House staff members who helped to fashion the Carter proposals, said that Congress probably will retain the last word on city spending.

"We want to move to minimize the federal authority, (but) it doesn't mean that the appropriations committees do not have final review," Dyke said. "The city should, however, be treated as much as possible like any other municipality. There are no other cities that have to go to Congress to have their budgets reviewed."

One possibility that may be considered, Dyke said, would be a removal of the appropriations committee's line-by-line veto authority and its replacement with a process similar to that affecting legislation enacted by the city government.

In the case of legislation, separate hearings are not held on each bill, as is now done with the budget. Instead, the city bills are allowed to lay over for Congressional review for 30 days. If Congress does not act to reject the legislation during that time, it becomes law.

There are at least two formidable obstacles that stand in the way of increased fiscal autonomy for the city.One is the Congress itself, where District appropriations sub-committee chairmen have traditionally guarded with jealousy their control over city spending.

Ironically, at least one leading city official believes that Sen. Patrick J. Leahy (D-Vt.), the Senate subcommittee chairman who opposed the proposed convention center, will be less reluctant to relinquish his power than his house counterpart, Rep. William Natcher (D-Ky.).

Natcher, who agreed to throw his support behind the 1973 home rule bill only if Congress maintained its fiscal stronghold, hasn't changed his mind since. But he is the fifth ranking Democratic member of the powerful Appropriations Committee, and in the three Congressional sessions between now and 1982, some city leaders hope he may move on to the greener pastures of a more prestigious sub-committee chairmanship.

The other major obstacle could be the city itself. Last year, Congress set up a special commission to oversee the city's financial operations after a private bookkeeping firm found that the city's books were so hopelessly inaccurate that they could not even be audited.

What one major city leader is now hoping, with all fingers and toes crossed, with all fingers and toes crossed, is that the commission's subsequent findings do not disclose something disastrous in the city's financial records.

The Carter administration's proposal is linked to a certain degree of fiscal responsibility on the part of the city, the official noted, and news of untold fiscal horrors in Washington's financial picture could be destroy the coveted budget autonomy plan.