Not quite, in practice they are pretty much equivalent but they're not Universal Turing Machines. For that they would need to have infinite memory. A Universal Turing Machine is a theoretical construct, it's not something that could actually exist.
Your definition is slightly wrong. In Alan Turing's own words:
It is possible to invent a single machine which can be used
to compute any computable sequence. If this machine I is
supplied with a tape on the beginning of which is written the
S.D of some computing machine M, {242} then I will compute
the same sequence as M.
(Alan Turing, 1936, On Computable Numbers, With An Application To The Entscheidungsproblem)
So it is no merely that the Universal Machine can read and execute programs, it is that it can read and executes programs to make itself equivalent to ANY other turing machine. Thus the Universal Turing machine is Turing Complete. As the amount of tape required to solve all computable programs is unbounded the Universal Turing machine must have an infinitely long tape.
It's usually worth going straight to the source where you can in CS, but much more so with Turing as his papers are so nice and easy to read :)
the tape only has to be $steps long, for $steps steps of computation, because that's the longest distance the head can travel at all.
any program that eventually halts, has a finite number of steps, so the band can be finite.
the only thing that requires an infinite band, is a non-halting program. and non-halting programs (unlike real-world servers and OSes) don't have a result as defined by the turing machine concept.
the only thing that requires an infinite band, is a non-halting program.
Sure, that's the only type of program that would use the whole tape, but given any tape of finite length, there's a (program, input) pair that needs a longer tape.
I'd argue that our computers are no universal turing machines, but they surely are close enough for everyones needs. Just like the ocean is infinite if you try to swim through it, even though a plane crosses it in a few hours.
