When playing a tune, do you make it your own?

yes.
you have to, else we hear the original artist in our heads and unconsciously imitate him/her.
Kinda like when we do a Irish accent singing Clancy brothers and Mary Black tunes or Dylan's tone creeps into our voices doing his stuff.

I will often "sorta learn it", meaning I listen to it a few times, get the words if it's a song and a rough idea of the melody if it's a tune. Then I try to work out a version of my own. Sometimes I'm fairly close to the original, and sometimes my version is very different. It's a lot of fun using a different instrument from the one used by your source for the tune/song. I don't like changing old time songs too much, but I can't imitate anyone else's voice, so my version is unique - not always better, but unique.

A few years ago I wrote a tune that has since been recorded by two different groups of musicians. I notice that they both use different changes from mine and each other's. I am not insulted in the least that they changed the chords some from the ones I used, because the ones they used were every bit as good as the ones I used to write it. I appreciate that they left the melody the same and am honoured that they chose to record my song. I try to do justice to any song I perform and hope that the composers or sources would be pleased with my version.

...sometimes, after playing someone's tune for years/decades!! you get the tab for it and find a section you have been doing completely wrong...and no one ever noticed.

Just saw that yesterday with Lady Stardust in G

D#m                    D#m(#7)
And he was awful nice
D#m7                  D#m6
Really quite out of sight


i have always done that progression in Am  (Am, Am(#7), Am7, D...and it still WORKS..


I just check some other verisons of tab for that song and they all show the chorus raised 1/2 step......
SO i took a listen to original album version..there is no discernible key change at chorus...

i am confuse

I like George Straits "Ocean Front Property" but when I sing the song here in Western Australia I change a few words:_ I've got an ocean front property in old Kalgoorlie, from my front porch you can see the sea.
For those not familiar with W A, Kalgoorlie is a town in the middle of what we refer to as 'The Goldfields' and as far away from the sea as Arizona of which George Strait sings. I do get people ask if I wrote the song but I do tell them that it was written by Hank Cochran, Dean Dillon and Royce Porter and was a hit for George Strait. I then say I've altered a few words to make it kinda Australian.