Scripture uses the term "flesh" in more than one way.
Matthew 26:41 -
"Watch and pray that you may not enter into temptation. The spirit indeed is willing, but the flesh is weak.”
John 1:14 - "
And the Word became flesh and dwelt among us, and we have seen his glory, glory as of the only Son from the Father, full of grace and truth."
John 3:6 - That which is born of the flesh is flesh, and that which is born of the Spirit is spirit.
Acts 2:26 -
"Therefore my heart was glad, and my tongue rejoiced; my flesh also will dwell in hope."
Substituting the term "sinful nature" for flesh in the above passages would not work. However, there are also passages that link our flesh to our sinful nature. But it's more than that. Here's what Martin Luther said about it:
“Thou must not understand ‘flesh,’ therefore, as though that only were ‘flesh’ which is connected with unchastity, but St. Paul uses ‘flesh’ of the whole man, body, and soul, reason, and all his faculties included, because all that is in him longs and strives after the flesh.”
Luther’s comments point out that “flesh” equates to affections and desires that run contrary to God, not only in the area of sexual activity, but in every area of life.
Susanna Wesley, mother of John and Charles Wesley described sin and the flesh like this:
"Whatever weakens your reasoning, impairs the tenderness of your conscience, obscures your sense of God, or takes away your relish for spiritual things, in short – if anything increases the authority and the power of the flesh over the Spirit, that to you becomes sin however good it is in itself.”
I find your reaction to people who use the term as descriptive of behavior to be what's the interesting part in your post. Could it be that since you didn't actually know what it meant that that was the driving force behind your not liking the term or thinking that people were using it as some empty phrase? I've personally never felt as you do when I hear people refer to behavior from the flesh. That phrase is certainly one of those "church phrases" that non-Christians who aren't familiar with is would be like "
huh? What in the world is he talking about?" But to those of us in the church, we should understand what it means.
Honestly, if you spend time in the Word, you will come across the term flesh quite often, so I'm surprised it's so new to you. It's used well over 100 times in the NT alone.