I think you are getting bogged down in interpretations and there is no blood running FROM her eye. When he talks about 'vertical' he means it's running down. He didn't mention blood running FROM her eye because no blood ran from her eye, it ran TO her eye from her nose and mouth.
If I'm getting bogged down in interpretation it's because it isn't clear to me what Dr Vanezis means:
"There was blood around her mouth which had run down her face in a vertical direction. In addition there was also a small amount of blood around her nose".
But the blood running down SC's cheeks towards her ears is the complete opposite direction of blood running down the length of June's leg and he refers to both as vertical.
The blood trails from or to the eye and from or to the mouth connect so one might have caused the other or not. The blood trials from the nostrils do not connect.
In any event like most aspects of the case I don't believe it is straightforward because we know SC sustained 2 gsw's and the pathological evidence suggests she was in a slightly upright position for both meaning some of the blood trails might have originated from the 1st gsw and others may have originated from the 2nd gsw when it is said she would have fallen back instantly. There's also the possibility blood may have pooled and then been released if she was moved.
Upshot is the most senior first respondent said her head was slightly raised against the bsc. Forensic scientist Dr Ismail said the exact thing based on blood stains and nightdress.