tl;dr: There is no evidence, to my knowledge, suggesting that Britain was the driving force behind rejecting use of the term genocide in the Security Council, however Britain and other member-states had reason to minimize the violence in Rwanda as less-than-genocide in order to avoid potential obligations under international law.

The quote from your linked source makes this question a bit hard since it provides neither a specific date nor resolution number. My best guess is that your source is referring to S/Res/912 as it is the only Security Council Resolution with regard to Rwanda in the latter half of April. S/Res/912 was adopted on April 21st, 1994 and essentially does a lot of condemning of actions in Rwanda and reduces both the size of the United Nations Assistance Mission to Rwanda (UNAMIR) and the scope of its mandate. I have never encountered another source claiming that Britain on the Security Council--manifested as Sir David Hannay, known as the "King of the Security Council" for his informal leadership among his peers--was the driving force behind rejecting the use of the term genocide in any Security Council resolutions, much less S/Res/912. In fact, Hannay wrote a cable on April 30th to Britain's Foreign and Commonwealth Office in which he discusses the presence of Rwanda's ambassador on the Security Council as the major obstacle to accomplishing much of anything with regard to the events in Rwanda that month and he specifically mentions Rwanda's ambassador insisting on not using the term genocide in a statement from the President of the Security Council (notably not a resolution, though potentially what your source intended to refer to):

As the evening progressed the Rwandan Ambassador's continued insistence on seeking deletion of the language on genocide and to the RGF responsibility for the bulk of the attacks on civilians... despite the efforts towards compromise made by the Presidency rather soured the atmosphere of the negotiations.

Nowhere in the Cable does Hannay indicate that this is beneficial for Britain. Indeed he conveys throughout the cable his dismay over how the Presidential statement was crafted. Since this diplomatic cable would have been quite confidential, there is no reason to believe that Hannay is lying about his own role in the deletion of the word genocide in the Presidential statement. From this we can infer that it is unlikely that Hannay was the one driving any decisions on earlier (or later, for that matter) resolutions to not use the term genocide. That does not mean, though, that Britain and other Security Council member-states did not have reason to deny or minimize what was happening in Rwanda.

The answer to your second question lies primarily in uncertainties surrounding intentionally broad language used for the Convention on the Prevention and Punishment of the Crime of Genocide. Passed by the United Nations General Assembly on December 9th, 1948 as General Assembly Resolution 260, the Genocide Convention was a clear reaction to the horrors of Nazi Germany's murder of 6 million Jews and 5 million other minorities in the Holocaust. Following 1945, most world leaders could agree that attempting to exterminate an ethnic or religious group violated natural rights and thus transcended borders, however what types of groups should receive protections under international law and what the rest of the world should do when another such extermination was attempted had little if any agreement between states. Indeed it is unlikely the Convention would have been passed or even written if not for the dogged efforts of Raphael Lemkin, a Jewish lawyer who lost most of his family members during the Holocaust.

The only way Lemkin was able to get Security Council member-states on board was to draft language that no member-state would view as a threat to their domestic affairs; for example, it is for this reason that we do not see any protections for political or ideological groups in the Genocide Convention, since the USSR was concerned that the repression of segments of their own population might then justify Western military action against it. Lemkin initially pushed for a collective agreement among states obligating them to come to the defense of any group facing extermination. He was emboldened in this effort by the North Atlantic Treaty talks of 1948-49 that revolved around common defense, however he quickly learned that states are much more willing to preemptively agree to defend a well-defined, state-level partner than doing to same for an unknown-until-attacked minority group in any part of the world.

Article I of the Convention, which is where the quote you pulled is itself pulling the phrase "prevent and punish". It is seemingly a simple and short affirmation of what the Convention is all about:

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.

When you break down the language of Article, though, it is clear that there is at least one small and one large loophole to implementing the convention's purpose of stopping, as quickly as possible, genocidal practices.

The small loophole is the lack of clarity on what it means to "prevent and punish". Consider an hypothetical affirmation between police departments that uses language similar to Article I:

The Contracting Law Enforcement Bodies confirm that murder, whether committed on public or private property, is a crime under provincial law which they undertake to prevent and to punish.

Without any further guidance, each police department could construct wildly different mechanisms for handling the prospect of murder and each would fulfill the affirmation. Department A might implement the death penalty for anyone found in possession of a deadly weapon, while Department B might require that every physically able adult possess a firearm on their person at all times and the responsibility to intervene with lethal force in acts of violence they witness. Each police department could argue that they have implemented policies to both prevent would-be murderers from carrying out their crimes and punish those who nonetheless do so. The affirmation doesn't require each police department to use the single best policy they can imagine, it only requires them to "prevent and punish". Similarly, states party to the Genocide Convention could conceive of different ways to prevent and punish potential genocidal actors. Upon reports of genocidal actions, State A might send a diplomatic team to work out a ceasefire, while State B might deploy ground troops to establish a DMZ between the base regions of the perpetrators and victims. Both are arguably valid attempts to prevent an incipient genocide even though the former will only prevent genocide if it succeeds entirely while the latter may limit the genocide even if State B fails to establish an impermeable DMZ. I refer to this as a small loophole because international norms and domestic laws may refine what it means to "prevent and punish" beyond the context of the Convention.

The large loophole, and the one that leads to the answer to your question, is that everything in Article I hinges on the presence of "genocide". Both the second and third phrases relate to the sentence's object, genocide, without exception. That means that if there is no potential for genocide, there can be no prevention of--and certainly no punishment for--genocide. As a result, it became understood among statesmen that the requirement to prevent and punish genocide itself required the acknowledgement that a genocide was occurring. By saying anything other than the genocide, Article I did not come into play. As such, states who do not want to be obligated by international law to intervene in a genocide in order to prevent or punish the perpetrators simply need not accept that a genocide is taking place. The Convention does not oblige states-parties in the event of crimes against humanity, for example. There is nothing in the Convention specifically stating that acknowledgement of genocide is a necessary prerequisite for the obligation to prevent and punish, but there was no precedent to contradict this belief either.

The events still unfolding in Yugoslavia/Bosnia and Herzegovina involving the United Nations Protection Force (UNPROFOR) and the very recent US failures in Somalia (of Black Hawk Down fame) made nearly all member-states liable to commit troops in the event of an increased UN deployment to prevent and punish a genocide very hesitant. So these states, in addition to their diplomatic actions, refused to acknowledge a genocide taking place in Rwanda to limit other Security Council member-states from taking steps toward forming a new and/or larger UN Peacekeeping force.