Who was the Escape Artist?

While Andrew is quite spot on in his description of both of them being Escape Artists at different points in the series (seeing their individual escapes from conviction as his answer and your question point out), I would see it from a different perspective. I would say that Will in his role as a barrister is the true Escape Artist. If the series shows us one thing with all these different cases, it's that there is always a way for a clever barrister to bend the law to his needs and help a criminal escape from conviction, no matter how pressing the evidence is. So not only the escaping criminals are Escape Artists, but even more so the lawyers that effectively facilitate their escape.

This is first emphasized in the dialogue near the beginning of Episode 1, when Will is unwillingly handed over the files about Foyle's case and his boss comments:

I can't say I blame them. If I was in their client's shoes I'd want your record on my side. If you are chained up in a safe at the bottom of a shark tank, you don't call for Paul Daniels...you call for Houdini.

Comparing Will to Houdini, and thus the ultimate Escape Artist, required to save the effectively doomed Foyle (seeing the pressing evidence against him). And it is this role as Escape Artist that finally takes its (at first moral, and later sadly real) toll from his life.

This is also to some degree reinforced by the BBC's introduction of the show saying

David Tennant plays Will Burton, a talented junior barrister who specialises in spiriting people out of tight legal corners, hence his nickname of 'The Escape Artist'.

(Together with Andrew's answer Will could even be seen as Escape Artist in multiple ways, especially in his last case, where he is accused and defender in unison.)

I took it to mean they were both the Escape Artist, in their own ways, but that really Liam was the true escape artist and his mantle effectively passed to Burton by the end of the series.

Whilst Burton was able to get criminals off and be a successful lawyer, thus ensuring their "escape", Foyle at the start appears to be the true Escape Artist. As you correctly point out, he is defended initially by Burton and escapes justice for a murder he committed.

Then, when Burton has no doubt whatsoever as to his guilt after finding his wife murdered, and after he comes down on him with everything he's got, he still escapes any justice due to Gardner's defence (despite even DNA evidence linking him to the crime).

Burton realises he can't win in the courtroom and takes matters into his own hands, leading to Foyle's death.

He effectively becomes the Escape Artist when he is charged with Foyle's death, brought to court, accused by Gardner of the perfect crime and yet still manages to escape justice.