In a way, the Bush Sr. story is a little easier to answer. Here he is, in October 1988:

The United States must acknowledge the attempted genocide of the Armenian people in the last years of the Ottoman Empire based on the testimony of survivors, scholars, and indeed our own representatives at the time ...

Here's Clinton in 1992:

... [t]he Genocide of 1915, years of communist dictatorship, and the devastating earthquake of 1988 have caused great suffering in Armenia during this century ...

Here's George W. Bush in 2000:

Armenians were subjected to a genocidal campaign that defies comprehension. Their travails should lead all decent people to remember and acknowledge the facts and lessons of an awful crime in a century of bloody crimes against humanity.

Obama said much the same too, but hopefully the pattern is clear: presidents are quite fine using the "g" word before they are elected. After, they're a bit more modest. It's important to note this is essentially a diplomatic dance, and there was certainly during-presidency acknowledgement. Here's George W. Bush again, April 24, 2004, Armenian Remembrance Day:

On this day, we pause in remembrance of one of the most horrible tragedies of the 20th century, the annihilation of as many as 1.5 million Armenians through forced exile and murder at the end of the Ottoman Empire. This terrible event remains a source of pain for people in Armenia and Turkey and for all those who believe in freedom, tolerance, and the dignity of every human life. I join with my fellow Americans and the Armenian community in the United States and around the world in mourning this loss of life.

It's essentially picking at toothpicks trying to claim "annihilation of as many as 1.5 million Armenians" isn't synonymous with genocide. But not using the "g" word was still enough to keep the diplomatic line toed.

What would not toe the line is an official recognition on the floor of Congress; for example, House Resolution 596 from 2000, "Affirmation of the United States Record on the Armenian Genocide Resolution", passed the House International Relations Committee; 24 "yes", 11 "no", 2 "present". It never made it to the floor by personal pressure from the State Department and President Clinton himself. For Bush Sr. in particular, there was a 1989 Senate proposal (sponsored by Bob Dole, later presidential candidate) that he opposed; the State Department claimed there was no contradiction between the positions, and that they hoped that Congress would "respond to the concerns of the Armenian people in a manner that does not gravely offend our vital ally and friend, the Republic of Turkey" and noted:

... [despite the] tragic suffering of the Armenian people and the need to commemorate the victims of the period 1915 to 1923, we are equally mindful of our close relationship and strong friendship with Turkey and of the differing views of how the terrible events of that period should properly be characterized.

The US has two military bases in Turkey, as well as other strategic interests; from that House Resolution 596 I mentioned earlier:

The United States and Turkey have worked closely over the past few years to arrange construction of oil and gas pipelines out of the Caucasus and Central Asia to ports on Turkey's coast, thereby expanding access to the two regions' vast energy reserves while avoiding Russian or Iranian control, and possible manipulation, of such pipelines. Turkey has also signed and implemented a military cooperation agreement with Israel, unprecedented for a Muslim state.

Reagan is probably the most interesting case, and despite his presidency being the most likely to get a full recognition of the Armenian genocide (for reasons I'll discuss momentarily) ended up eventually following the same pattern, still opposing official recognition from Congress.

...

Reagan was California governor from 1966 to 1974; California happens to have a large and strong Armenian diaspora. He gave a 15-minute speech at one of the Remembrance Days:

Today, I humbly bow in memory of the Armenian martyrs, who died in the name of freedom at the hands of Turkish perpetrators of genocide.

During his governorship, Turkey had invaded Cyprus -- the details are too much a sidetrack for this answer, but the end result was an arms embargo from the US on Turkey. While the embargo was lifted several years later, relations remained frosty in the 1970s.

March 1980 marked a turning point with DECA (the Defense and Economic Cooperation Agreement); Turkey could get modern military tech from the US in exchange for access to bases.

When Reagan became president a year later, he used the "g" word again in April at a Holocaust Memorial event:

The millions of deaths, the gas chambers, the inhuman crematoria, and the thousands of people who somehow survived with lifetime scars are all now part of the conscience of history. Forever must we remember just how precious is civilization, how important is liberty, and how heroic is the human spirit.

Like the genocide of the Armenians before it, and the genocide of the Cambodians which followed it -- and like too many other such persecutions of too many other peoples -- the lessons of the Holocaust must never be forgotten.

Actual effect at the time? Not much. Some Turkish media mentioned it, but Reagan never got an official backlash; Turkey needed DECA badly enough it didn't want to raise an official fuss through above-board channels, and the mention was incidental enough it didn't present a full-on challenge that needed saving face.

There was almost certainly a back-channel fuss, evidenced by the US State Department putting out a statement out the next year claiming the "genocide" matter was controversial and the closes the US has come to a denialist position (note: this was later withdrawn).

It's faintly possible Reagan might have pushed his luck farther; he had the opportunity to in 1985 and 1987 when Armenian Genocide Resolutions were brought to Congress, and Reagan repeated the same pattern of being against them.

He did still do something major, and encouraged adoption of the UN Genocide Convention. This had been created all the way back in 1948, and was never signed into law until Reagan pushed for its passage in 1986 (a modified version was signed in 1988).

The Contracting Parties confirm that genocide, whether committed in time of peace or in time of war, is a crime under international law which they undertake to prevent and to punish.

Certainly, one could read the "universal genocide bill" as including Armenians; the revival of a 40-year old bill certainly meant genocide was a personal concern of Reagan. So why reject Congress on both opportunities to name the Armenian genocide specifically? There's not enough evidence to say definitively, but I suspect it has to do with Armenian terrorism.

We'll need to back up a little, again--

From 1973 to 1988, there was the Armenian Secret Army for the Liberation of Armenia (ASALA), the Justice Commandos of the Armenian Genocide (JCAG), and the Armenian Revolutionary Army (ARA).

They were responsible for bombings and assassinations targeting Turks. Throughout their existence they killed 90 people and wounded hundreds. Their demands were 1.) recognition of Armenian genocide 2.) reparations for genocide survivors and 3.) return of lands taken during the genocide.

Obviously, they never made it past demand #1, but not for a lack of trying. To some extent, their efforts could blur together with the other militant leftist groups from the 70s, but their actions in the 80s in particular served to alienate potential Western allies at a time they could have been sympathetic.

In the summer of 1982 (the year after Reagan made his genocide comment) there were assassinations in Ottawa, Canada and Burgas, Bulgaria. The US and Turkey formed an U.S.-Turkish Joint Group on Armenian Terrorism. Five ARF terrorists in the US were caught with bomb materials by the FBI.

Three events in particular loomed over July 1983:

July 14, Brussels: the assassination of Durson Aksoy, Administrative Attache to the Turkish Embassy, shot twice in the head. All three of the organizations I mentioned (ASALA, JCAG, ARA) claimed credit.

July 15 (the next day!), Paris: a suitcase bomb detonates at the Turkish Airlines check-in. Eight were killed (including an American and four French citizens), sixty wounded. The bomb went off early and was intended for the flight (which would have killed all 167).

July 27 (the same month): five terrorists from the ARA attack the Turkish Embassy in Lisbon. The forces are repelled by the Portuguese, but 7 are killed. The ARA claims their "brothers would be avenged" and threatened the Portuguese Prime Minister.

These events on European soil in quick succession were utterly shocking. Reagan himself:

No real or imagined grievance could possibly justify these modern day horrors.

It is difficult to imagine, given his anti-terrorism stance elsewhere, of pushing hard on a cause that even aligned incidentally with terrorists.

To summarize:

1.) Reagan had sympathy and connection to the Armenian diaspora going back to his time as governor.

2.) His mention of genocide was incidental and there was a crucial agreement involving military technology the Turks wanted to preserve. Based on the State Department's statement a year later, there was still back-channel protest.

3.) Reagan might have potentially pushed farther but Armenian terrorism undermined potential sympathy amongst the West.

4.) Later presidents used the "g" word but before their presidency, yet during it they still indicated in more diplomatic language (like "annihilation") roughly the same thing.

Reagan's 1981 speech, incidentally, was written by Kenneth Khachigian, whose father was a survivor of the genocide.