Hi LJ,
I too craved a Strawberry bag recently, before the December holidays. I was wary of it having too much warm red in it because warm colors clash with both my skintone and the wardrobe of cool colors I've collected to complement my coloring (and because I love cool colors anyway).
I bought a Belvedere collection bag in Strawberry. It is thankfully not too warm, but in some lights more than others it leans more red than pink, and vice versa.
When I had to tell someone on the phone how to quickly identify me on the street, I said, "I'm holding a red handbag." I actually had to pause before deciding to say Red instead of "bright pink." The average person--especially someone like many men, who haven't been privy to a subculture of fashion, crafts, cosmetics, or other color hobbies--associates "pink" with a baby pink, carnation pink, etc.
Ultimately, I believe many people would see the Strawberry (Belvedere) and say, "the red handbag," even though the touch of pink in it means it might compete with rather than complement a carrier's true red scarf or shoes.