Push Poll this evening- turned out to be Republican.

[No Vested Interest]

4. The poll on my phone limited my answers by virtue of having me push numbers on the phone

to indicate agreement to their set answers.
It was not conducted by a live person, but a recorded questioner.
It purported to be a "public opinion poll".
I considered the first few questions were to put me at ease with the poll.
There was no alternative to what was (for me at least) the final question - For whom would I vote in the Republican primary?

The example you cite is certainly a type of push poll, but not the only one.
Whoever commissioned the poll or whoever wrote the wording was looking for certain responses, not conducting an honest survey of public opinion. That qualifies as a push poll.