Splicing squares on a Turing Machine finite tape
Yes, it has the same computing power as an ordinary TM.


It's a standard exercise in computation theory/complexity theory textbooks that the ability to insert characters into the tape (and delete them from it) can be simulated by a standard Turing machine with a quadratic loss of efficiency. For example, this is Exercise&nbsp;2.8.8 of Papadimitriou's book (Computational Complexity, Addison Wesley, 1994); he doesn't give a name to this extension of the standard Turing machine or attribute it to anyone which suggests to me that it's folklore &ndash; he's pretty good at citing and attributing.
Since you can insert characters at will, reaching the end of the initially finite tape makes no difference: you can just use insertion to extend the tape. A halting computation will only need to do this a finite number of times.
The infinite tape of an ordinary TM is just a definitional convenience: it suffices to have a finite tape that gets extended any time the head reaches the end.