[hansvm]

The usual difference is just predicate ordering -- (1) for every program there exists a tape big enough vs (2) there exists a tape big enough for every program. In the first case, each individual (valid) program can get by with a tape of _some_ fixed length, but there's no bound on how big that requisite length might be. In the second case, since the tape requirements can be arbitrarily high you would need a legitimately infinite tape to service all possible programs.

IMO the given example muddies the waters a bit by conflating the conceptual tape a given machine is running on (which might be infinite for non-halting programs) with the physical tape you've used so far (for which it suffices for such tape to be finite at any fixed moment in time, though the amount needed might grow unboundedly as time increases).

[codeflo]

Note that tape usage typically depends on the input, so I would distinguish programs and computations (program + input).