The problem has to do with a quirk in the png format, the way InDesign handles image sizes and a missing step in your workflow.
The size in pixels per inch of an image is determined by its metadata. A 1000x1000 pixel image can be 300 ppi or 72 ppi or even 2 ppi and it will still be the same 1000x1000 pixels in size. The only difference is the number hidden in the internal metatdata of the image. PNG, however, does not contain any ppi information (it also contains no color profile information, which can land you in trouble if you need color accuracy).
When you import an image, InDesign determines the size of the image (in inches or mm) based on the ppi number in its metadata. Since PNG has no such number, InDesign uses its default: 72 ppi. That means a 720 pixel image will be 10 inches wide if you place it by simply clicking the mouse where you want it to go. At 100 ppi, the image would be 7.2 inches wide, and so on.
This brings up the missing step in your workflow: don't just click. Click and drag to place an image. That way, you determine the size of the image on the page, not the metadata in the image.
Your PSD is probably 300 ppi or greater. Placed in InDesign, it will be sized based on the ppi information in the PSD: pixels/ppi = size. Your cropped PNG has no ppi metadata, so it will be sized using pixels/72 = whatever that happens to be.
In general, don't use PNG as an image format for print unless you know for certain that your document's RGB profile is sRGB and the original image was sRGB before you exported as a PNG. PNG was developed as an image format for the web specifically, so it didn't require a ppi number or a color profile.