The poem sets up aesthetic appreciation—peonies like Chinese porcelain, wonderful, glowing—then swerves hard at "But, my Dear." That comma after "But" is crucial. It's not "but my dear friend" (address). It's "but, [listen] my Dear" (intimacy + correction). The speaker is talking to someone who might admire the peonies, and she's saying: *you're looking at the wrong thing*.
Larkspur vs. peonies is common vs. exotic, native vs. imported, blue vs. pink. Larkspur is a cottage garden flower, nothing special. But it "swings windily against my heart"—that physical verb, swings, makes the flower hit her chest. Not metaphorically touches. Literally swings in wind and strikes. The peonies glow at a distance like art objects. The larkspur makes contact.
The cricket is the third element: sound, not sight. After two flowers described by color, we get an auditory memory. Crickets mean late summer, dusk, the end of things. The poem moves from visual beauty (peonies) to physical sensation (larkspur swinging) to sound (cricket)—and then stops. No resolution. Just the afterglow of "Other Summers" that won't fade.