it really depends on the 1. weather 2. your route 3. your skill level. Check out these excellent weather conditions resources:
I went there two times (without a guide) last year, once in July and once in September. July was much colder weather, with strong winds. My partner made it to the top wearing only heavy fleece as his insulation, bus said it was chilly on top. September was very warm, actually too warm.
My clothing was:
Icebreaker lightweight base layer
Light fleece/or MH Powerstretch Zip Top
Montbell UL Down Inner Parka
GoLite Phantom (goretex paclite jacket)
Patagonia Capeline 1 tights
Mammut Nimba softshells (they have microfleece liner)
I had an extra fleece in my pack, which i never used. I only put on my parka when i stopped to rest.
I also had a neck gaitor, a fleece hat, googles (!!), and RBH mitts and lightweight powerstretch gloves. On my feet, i had stiff Scarpa boots, not exactly mountaineering boots but very similar, and much less expensive.
The fact is , if you know the weather is stable and are not trying to "fit in" the climb into your schedule, summer Rainier climbing can be done pretty light. Denali list is an overkill, as it assumes minus 20 degree temps. Plus if you are going the DC route, which you are i 'm assuming, and during a weekend, it's like a trekking highway. Most of the people i met over on DC could be technically considered as trekkers, but came prepared as they were climbing an 8,000 meter peak.
Regrets? i am considering getting a loftier down jacket, perhaps something with at least 5 ounces of fill weight. June 1st is still very cold month for Rainier, and one that could present with an unexpected weather. You may want to get a good down layer. Positives: less open crevasses, more direct route, and less people.
On my second attempt, i saw this guy doing a triple record from Paradise parking to the summit via the DC route, in his trekking shoes, wearing a day camelback, very thin layers, and a pair of trekking poles. Of course the weather was excellent, and this dude was at some point a denali speed ascent record setter.