Berkeley Italian for TONIGHT
Rivoli, while IMO the best restaurant on your list, is not traditional Italian. I'd describe it as "continental" - there are strong French, Italian and Spanish influences. Also, you should probably expect to spend more than 125/2 - it could theoretically be done, but you have to to split at least one course and order one of the cheapest wines on the list.
Corso is a Rivoli off shoot, very simple, rustic Italian food, quite good and affordable.
La Siciliana is great as well, especially the delicious pastas. I prefer to visit with parties of 4-6 because the best way to experience it is to order family style and try a bunch of things.
Adagia and Venezia have recently received less than stellar reviews, but I can't personally comment.
Unless I'm thinking of a totally different place, La Med is not Italian at all.
From your list I would say Corso or La Siciliana are the best choices considering your requirements.
I'm currently having a bit of a love affair with Corso. Although even after three/four months they still haven't fixed all their opening night problems. (Utterly lame waiter on one visit, frito "misto" consisting of only calamari crumbs on another, they've still got the frigging -television- in there). So we may have to have The Talk soon.
However, the grilled whole fish (usually branzino but it was mackerel one night) is probably the most delicious thing available in a restaurant in Berkeley at the moment. The tagliatelle is very, very good. The chicken breast roasted in a pan of butter is wonderful. Interesting selection of wines almost-but-not-quite bargain priced and available in a variety of sizes ranging from taste, glass, up through half and full bottle maybe all the way to jeroboam (well, maybe not, but I like to say jeroboam. "jeroboam"). Pizzas are just average.
It seems to cost about $85 total everything for two whenever I'm there.
Oh, one more problem -- it can be stupidly loud in there. If you yell to your waitperson, "HEY, IT"S REALLY LOUD IN HERE!" she will yell back, "YES, WE"RE TRYING TO DO SOMETHING ABOUT THAT!" Except they're not. Underneath all the crowd noise they have the stereo turned up REAL LOUD TOO. Eventually I hope they outgrow the loud cliche. Since it rarely works as planned.
Anyway, that's where I'd go.