When it comes down to it, InDesign doesn't embed any fonts — even OpenType ones. If you want to illustrate that point, just use a couple of downloaded Adobe Fonts in your document, close the file, turn off the online service, and reopen it to see what happens.
When exporting to Adobe PDF, you have the option to download fonts, either the entire font or subsetted characters of it, depending on their use in a given InDesign document and the settings of your native InDesign creator (preferred) or external tool like Acrobat Distiller or other PDF generators.There are outside factors there too, depending on who provides the fonts, how they're provided and the assigned rights for using them. But setting those aside, when you're talking about "embedding" fonts, you're usually talking about embedding them in PDFs. And there you can embed OpenType, Type 1 and/or TrueType fonts. Short of licensing issues, the only fonts I know of that cannot be embedded in PDFs are ancient, Postscript Type 3 fonts which haven't been used in production systems for roughly 30 years and should've been banished long ago.
Mike's steering you right when he suggests you should use OpenType fonts wherever possible. They're your best bet to consistently get the results you expect when moving InDesign documents and PDFs from system to system. Also, the best option for ensuring embedding and portability for your InDesign documents/PDF files anywhere you need them to go is to use resident fonts that are installed on your system.
Hope this helps,
Randy
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