First week of August in Scotland : What kind of clothes to pack?

I was in Edinburgh in the first week of August a couple of years ago, and I got a quite wide array of weather conditions, often on the same day: cloudy, then sunny, then cloudy again with rain, and so on.
Regarding clothes, I usually went around the city with jeans, sneakers, a t-shirt, a light sweater and a light waterproof jacket for the rain. The temperature was never too cold, so when the sun came out I took off both sweater and jacket and walked around in my t-shirt, and I wasn't cold at all.

I live in Edinburgh and in August I'm sweating if I go out wearing jeans in the daytime, especially if it's sunny but often even if not. I'd advise wearing thin/loose trousers and a T-shirt, then take several layers (thin jumper and maybe a waterproof of some kind) so you can layer up if the wind picks up or it begins to rain.

A light waterproof jacket is a better idea than an umbrella since Edinburgh is often windy - which is also the reason why I recommended thin trousers rather than shorts, and a thin jumper as an extra layer. Even if the air temperature isn't too cold there might be a bit of a breeze, and in summer you're more likely to be chilled by the wind than needing huge thick jumpers.

Regarding packing, I would say take as many different options as possible. Then check the weather forecast the day before (or in the morning) to decide what to wear on the day. There are plenty of smartphone apps that can do this for you, or you can use the internet/TV in your hotel if you have such facilities. The BBC news is at 6pm and 10pm (and has been that way for years so is unlikely to change) and they have weather about 20-25 minutes in.

This is a place that has lots of different weathers from day to day, so bring a small light jacket, and jeans etc,

It might be warm, but it won't be roasting weather