How to properly connect an old record player output (spade connector, possibly amplified) to an RCA tape deck with built in amplifier?
It's important for you to figure out whether the output that you refer to is a line level signal (i.e. just conveys information, not power), or an amplified signal (designed to drive speakers directly). If it's a line level signal, then yes, you can plug that into a tape deck with an amplifier. If it's an amplified/driven signal, you're likely to burn out your tape deck's input.

To figure this out, you should look at the manual if you can find it online. Is that output meant to drive speakers? If so, it's already amplified. It should be passed to speakers, not another amplifier.

As far as what those connectors are called, I'd just call them lugs. To connect RCA cables to it, buy an RCA cable (RCA on both ends), cut it in half and strip the insulation off near where you cut. You now have two RCA to wire "adapters". Use a nut/bolt or similar to attach them to the horseshoes.