Having read up a bit more, I think I can explain a few of the apparent discrepancies
These links are from Neil Berrimans website. Annoyingly there are some pages missing but they still give some extra detail.
www.lordlucanthetruth.com
Confidential+2002+Lucan+Police+report.pdf (squarespace.com)
ORIGINAL+LADY+LUCAN+STATEMENT.pdf (squarespace.com)
The tea cups and saucers - According to Lady Lucan's statement, Ms Rivett had taken a dirty set downstairs with her to make a fresh pot. That makes sense and clears that up.
According to police analysis, the blood in the basement was mostly Ms Rivetts, apart from a couple of samples from the kitchen floor and the mailbag which matched with Lady Lucan. This is a bit strange unless things didn't quite happen as Lady Lucan remembered - very possible since she suffered some pretty severe injuries to the head.
Lady Lucan had five lacerations to the scull (sic throughout) and the forehead. They were deep and jagged and in places as deep as the scull. It is also possible that these wounds were caused by the weapon found at the scene. If Lady Lucan had received these wounds to the rear of the head, they may have been fatal. She also had lacerations on the inside of her mouth.
Now, even if, for the sake of fairness we exclude Lady Lucan's testimony, Lord Lucan had been acting increasingly unhinged and terrorising not only his wife but the nannies. Hiring private detectives to follow them, making weird phone calls and bringing pills to a pharmacist to have them identified (what was that about?). This is all from the Wikipedia page. Astonishingly, he did indeed have a key to the house which is mentioned in the 2002 police report.
Here is what may have happened. Lord Lucan let himself into the house and removed the bulb from the basement light. Ms Rivett comes down with the teacups. He wasn't expecting her to be there but never mind, he is "Lucky" after all and she deserves to be killed too due to the influence she has on his children. He hears Lady Lucan coming down from upstairs and runs up to ambush her. He murders her too (he thinks), then goes back downstairs to continue dealing with Ms Rivett's body, contaminating the scene with Lady Lucan's blood. He goes back upstairs only to find that Lady Lucan isn't as dead as he thought. Maybe it is a bit harder to murder someone who is sitting up and talking to you, who knows? Anyway, he is all murdered out at this point and she survives. His words to Lady Lucan were:-
I said Where's Sandra? And at first he said "she's gone out". I persisted in asking this question and suddenly he said "I've killed her, she came down first, if it had been you, you would have got it."
It sounds as if he knew perfectly well who he was murdering and didn't much mind.
The other discrepancies are a parking attendant at the Clermont club claiming to have seen him at 9pm (pretty much the exact time of the murder) in his Mercedes. Lucan asked him who was present and then drove off. The Mercedes was found after the murder with a cool engine and a flat battery. Lucan could have easily bribed the parking attendant to say that.
This only leaves the second lead pipe which I still find a bit strange. Though who knows, maybe he made one pipe up and put it in the boot, forgot it was there and made up another? As previously noted, this is not an ordered mind.