Obviously they will need to be installed on the New mac to appear correctly. I find you sometimes need to relaunch Illustrator (CS2-6) to get it to recognise fonts after installing them.
The * next to the font indicates it's not installed or not recognised as being installed.
Have you migrated all your fonts from your old mac to the new one?
How are you managing fonts (Font book, or some third party software?)
Some similar here. Perhaps the full set are from the full family, here…
I think that is Edition, with some interesting scaling to make the copy fit a block…
Both named fonts download OK for me, so as you seem to suspect, your problem would appear to be local.
The name of some fonts you have been trying to download might help, so it can be checked to see if the problem is with your system, or a broken link.