The first thing to realize is that in Europe skiing is basically a winter sport. I might just be stating the obvious but it seems that this question is surrounded by confusion and unstated assumptions. For example, most if not all of the places listed in your link will have no snow at all and a relatively warm weather so the reason this site and others don't provide a “definitive answer” is that it would completely obvious to their customers that you can't ski there in the summer. Instead, what you will find in these resorts are summer activities like hiking, paragliding, rafting, open-air swimming pools, crossword making, pottery workshops, yoga classes, etc.
The main possibilities to actually ski in the summer in Europe are listed in the link provided by @Karlson (Ski Resorts during the Northern Hemisphere's Summer) but it's really a niche activity, limited to a handful of glaciers plus some ersatz skiing (indoor skiing, grass skiing, skating…) that really aren't much like the real thing.
In any case, winter or summer, skiing is quite expensive. I don't think you will find many cheap accommodations in typical mountain resorts (what most people do is rent a house with a group of friends, it's usually cheaper than a hotel but can still be expensive). You also have to factor in the cost of the ski pass (the right to use all the lifts and the like to repeatedly bring you up the mountain), renting the equipment, insurance, possibly a course or two. Without some equipment, a group of friends or a local contact, I think it is next to impossible to go skiing for a week for 1000 €, especially for a beginner, certainly not if that amount should also cover the transport to get there in the first place (you can manage a comfortable week-end with this budget but of course that's a lot of money to spend at once if your resources are limited).
If the reason you are considering this is that you happen to be in Europe, can't wait for the winter and don't expect to get another occasion to go the Alps soon, then you should definitely go there but I think that focusing on skiing is a waste of your precious time and money. You can for example go hiking, it's completely different from skying but very satisfying in its own way and a great way to experience the beautiful landscape and discover the mountains. If you want some thrill, you can also do something like paragliding. If you really want to see some snow, a trip with something like the Aiguille du midi cable car could be added to your plan as well (it's pricey but still much less than ski lessons and equipment).