Aramaic words
In Aramaic, the word traditionally translated as ‘cross’ is a word which only means a structure that’s ‘erect’ or ‘straight up’, or ‘high’. It can refer to something used for either cruxifiction or hanging.
We translate it as ‘upright’ while adding a [translator note] to clarify what it means. So our translations from the Aramaic books usually say ‘the upright [execution stake]’ or something like that.
We don’t say ‘cross’ because that’s not what the Aramaic word means. Also, a cross was just one of many different shapes used by the Romans on such uprights. The Bible text does not specify the exact shape of the upright used to execute Jesus, only that an upright was used. The fact that the severely beaten Jesus was able to carry part of this upright (at least for a little while) does suggest that he was carrying a cross-piece rather than the main upright pole. However, this doesn’t tell us about what shape it was used to make at the execution site.
So, while the idea of the traditional cross-shape could easily be correct, the Bible doesn’t indicate wether its shape ended up resembling a +, t, T, or something else. We only have tradition to go on.
Greek words
There are two Greek words used to describe the instrument of Jesus’ execution: xulon and stauros.
Like the Aramaic word, neither specifically mean a cross-shape.
Xulon just means a piece of wood. It may mean something as large as a tree trunk (e.g. the tree of life in Revelation is a xulon), or as small as a wooden club (e.g. the men who came to arrest Jesus were armed with xulon, or clubs). See Strong’s Concordance for more information.
>Stauros just means a pole or a piece of lumber, but especially a pointed stake. This was the usage of other ancient Greek writers, including Homer, Herodotus, Thucydides, and Xenophon. See Strong’s Concordance for more information.
Translating these words as ‘cross’ is merely a tradition that appeared in later times. As stated above, the Bible does not contain a description of a pole with a cross piece (making a cross), it’s only inferred from the context, tradition, and Roman sources.
Therefore, our translation of the Greek books uses ‘tree’ for xulon and ‘pole’ for stauros because that is what the manuscripts say. Also, staurotheto (which other Bibles translate as crucified) is shown as impaled (put on a pole or stake), because that’s what it actually means.
Are we saying that Jesus didn’t die on a cross? Not necessarily. There’s quite a bit of historical evidence to indicate that he likely did (see the Wikipedia entry, Instrument of Jesus’ crucifixion). The Romans used a variety of different shapes, and the cross-shape was frequently one of them; but that doesn’t give us permission to insert a description into the Bible text that wasn’t there originally.