Accepting Murasaki's predilection to show her own cleverness, it is no surprise that
she would toss in technical terms from a difficult game. But this is a little bit
more than showing off because she uses them accurately, and given the nature of the
terms - not ones beginners would use - there is a very strong implication that she was
herself a decent player.
The first term is kechi 
. The term has been out of use for centuries, but you can
still find it defined in large general works such as the Nihon Kokugo Daijiten (Shogakukan, 1974).
Slightly incorrectly, it has: Igo no shuubansen de, mada kimenai me wo tsumeyoseru koto
(At the endgame stage in go, playing the yose at the points not yet decided). It is
slightly incorrect because the yose is not limited to the endgame stage. Tsumeyoseru
implies yose though the tsume- prefix is perhaps misleading and could make people
think too much of "filling in" as dame wo tsumeru. The real point of the verb yoseru
is that it refers to the process where opposing territories approach each other, or are
drawn together or bind together. There is some filling in involved in that process
but it is secondary (hence tsumeyoseru rather than yosetsumeru).
Specialised works on Tale of Genji should give more help. The Genji Monogatari Jiten - Tale of Genji Encyclopaedia by Ikeda Kikan (Tokyodo, 1960) says for kechi: "One example. Technical term in go. When the game is ending, disposing of the points which are not territory for either side ( dame)." Unfortunately this is not really correct either, but it does go on to quote a usage from another old work, Kakaisho, with the gloss katame. Suddenly we are on firmer ground. We know that this term meant yose in Edo times. It even appears in the Gokyo Seimyo with the same character as kechi.
The base meaning of katame is exactly what we would expect for a term describing this phase of the game: hardening or consolidating. It is the process where the boundaries of territories are consolidated, or bound together. It is not the final stage where neutral points are filled in prior to counting.
In general it is difficult to talk confidently about rare words in old texts, but we can speak with some confidence about this one, because it was rather better defined in a go book that appeared in 1199, a relatively short time after Murasaki died. This was the Igo Shiki by Genson. The original is lost but the content has been preserved in the huge compendium of ancient texts that were collected in the Edo period into the Kou-Gunsho Ruiji, in this case Part 360 devoted to Pastimes.
Genson, who was probably also the original source (model even?) for the famous story of Kanren the go-playing priest (on the GoGoD CD), defines kechi as follows: "After roughly the halfway stage of the game, we focus on the kechi [the yose] and to take the lead in the kechi means not allowing the opponent to have sente for even one move... At the counting up stage when determining the result, if you have played the kechi well, you can catch up even if you were losing by 20 points."
Straightaway we can tell we are listening to a pretty advanced player, at the very least a high-dan amateur in modern terms. This speaks volumes for the general strength of go at that time (confirmed by game records from the roughly contemporaneous game records of Song China), so we need not be shy of thinking that Murasaki herself could have been a very creditable player just through being accurately familiar with such terminology.
Actually, it is also Genson to whom we owe the advice to count three times during a game: in the opening, middle game and endgame (kechi).
Ji 
is the next term.
We lack a contemporaneous explanation for this in Japanese, though the famous
list of 32 go terms from Tang China has it, and these were almost certainly
known in Japan. There is after all a matter-of-fact allusion to the Chinese poet
Bai Juyi's famous line about go in Genji (Yadorigi chapter) and much else from
the Tang dynasty. But, in any case, throughout
classical times and even down to modern times this has been the character for seki.
This too is defined in the Genji Encyclopaedia, with passable accuracy this time as "Technical
term in go. When stones are in contact and have either one eye or no eye but
both sides are alive." Oddly it makes no reference to the term seki.
Ko presents no problems (Genson mentions it, too) in oriental texts, so there is no need to discuss it further.
If my version if this text is accepted, it is worth deeper scrutiny. After the "wait!" the calm lady is carefully examining the yose and asking the other to be patient while she does so. The real point of the comment is to contrast calmness with excitability, and it can be fairly said that the other translations convey this message.
Then the action is telescoped slightly and we are to assume they have counted up. This telescoping is typical of the book (and the fact that Murasaki has mastered such stylistic techniques is one reason the work is regarded as a masterpiece). The lady who lost has forty points, which would nearly always be a losing total in a 19x19 game - another touch of intimate knowledge by Murasaki. But this total is interesting in another way. It seems to imply the "Japanese" style of counting, which we know was really the ancient Chinese (Tang) way of counting. A 40-point total would be almost nonsensical under modern Chinese counting. It is tempting to note also that there is no reference to group tax and to claim that Murasaki would have been bound to include such a term, but that is almost certainly making the text bear too much weight.
We do, though, have to deal with the possibility that the counting referred to some form of wagering. In normal gambling go, the payment depends on the margin of victory, rounded off in tens.
But there is no reason from the text to assume any kind of wagering involving counting in Genji, here or elsewhere. In the Takekawa chapter, the end of a game is marked not by a settling of debts but by a joke about playing a Korean melody regularly used to signal a victory in other contests. And in the Yadorigi chapter, a royal personage tells a counsellor that he has something to wager on go, but it turns out to be a flower, and it is decided on the outcome of three games.