I'll differ from most current thinking on this and say I do think there is an element of dominance here. I think your dog is coming into a sense of her adult self and is developing a sense that she is more powerful than your other dog and can subsequently control situations to her advantage. I don't think this will diminish if the status quo is maintained as she is constantly 'winning' these encounters and this is cementing her idea that she can control all future scenarios in a similar way. Add to the fact that she is only 9 months old and still developing so as she becomes more adult she will likely consolidate her position.
Now to my mind this does not mean that you acting in an overly punitive or dominant way towards Willow will help anything, nor does it mean that she will always act so aggressively, as it may be that, in time, she finds that a look to Tilly is enough to let her know she is not allowed to sit anywhere near you when you are eating for example, and Tilly may decide enough is enough and not even attempt to go near anyway as she has had enough of the aggressive displays.
I think there is a trend now to interpret a lot of behaviour as fear-based, and there may be something in this to a degree - fear of losing out on food or attention - but that doesn't mean that a dog is not a bully, and that does not mean they are displaying an appropriate response to fear. Two dogs may both feel fear but have completely different behavioural manifestations of this - and we still have to control when the behavioural manifestation is harmful. As a parallel you could say (depending on your politics) that Donald Trump uses the rhetoric he does because he is fearful and anxious - of foreign people coming into his country, of religions that are not his own, or you could actually just say he is a bit of a bully and trying to control people, resources and situations to his own advantage. Fearful behaviour, to me, does not have the best explanatory power or potential for successful resolution for Trump's rhetoric and actions and it does not always have the best explanatory power for all dogs behaviour (although I certainly do not dismiss it and recognise it as a clear factor in my own dog's behaviour at times.)
You are still the ones in control here and it must be very hard when Tilly is having a hard time out of this. When your food is involved Willow is motivated by her own access to it and prioritising that over Tilly's access - but you are the ones that ultimately control any access at all. So you can maybe (if you aren't already) try ensuring titbits are never available to either dog under any circumstances and never allowing there to be a situation where the dogs are potentially vying for access to food. Similarly you have ultimate control of the attention you give the dogs and you can make it clear that you don't find it acceptable if Willow snaps at Tilly if she is receiving attention from yourselves - you can possibly keep her on a light long lead in the house to remove her from fights. I am probably in danger of being overly simplistic about your situation here and apologies if this is the case - this sounds a very hard situation for you all and really tricky to rectify especially as your dogs probably have a sense of 'this is between us'!
