The original question was quite different making my initial answer irrelevant. (It originally asked what this -tate means and assumed that it was past tense -ta plus -te.)
In what way is a version of the conjunctive form (連用形?) ?
All verbs have a 連用形.
All of the examples that you gave are 一段 verbs, so the 連用形 looks the same: tore-ru --> tore, deki-ru --> deki, umare-ru --> umare.
If you expand that to include 五段 verbs such as yom-u, you will see that the 連用形 is yomi.
If you are not familiar with the grammatical vocabulary, the easiest way to remember this is that 連用形 is the form that -masu attaches to. So, tore-ru --> tore-masu, deki-ru --> deki-masu, yom-u --> yom-i-masu, umare-ru --> umare-masu.
Just like -masu, -tate attaches to the 連用形 of verbs.