First point; evangelicals "pick up their cross" to follow Jesus exactly the same way as all Christians are to do. There is absolutely no need to differentiate evangelical Christians from all other Christians unless you have some evidence to present that non-evangelicals have a distinct interpretation on such a verse.
Second point, I agree with the comment made by Steven L. Hitchcock that there is no warrant for the word 'daily' in any of the three verses where Jesus makes this statement.
Third point, I am happy to go along with your request for a "working definition, or a practical definition, or a practical application for take (up) your cross". I am an evangelical Christian myself, and I recall a lecture given by a professor of the Free Church Bible College in Edinburgh on this very text. He clearly said that the text does not mean enduring some physical suffering or hardship in daily life. He objected to people claiming that their arthritis (or migraines etc) were their personal 'cross' they had to endure. He made the point that it has to be something endured for the sake of now being a disciple of Christ that the text speaks of. Nobody suffers arthritis or migraine because they have become a Christian!
Another point is that, when Jesus said this, he had not yet gone to the cross himself. His disciples, whom he was sending out, were being warned that it was costly to follow Jesus. But the cost was understood by them as being figuratively akin to the shame and horror of Roman crucifixion, with which they were well familiar. In other words, they were not to look for glory and acclaim for being Christians. On the contrary, they could expect shame and suffering. Were they prepared to take that up, for Jesus' sake?
Jesus had just said that their love for him had to surpass natural affection for family, so they might well grasp that, in the next breath, to take up their cross to follow him could actually cause their family to despise them. Would they be prepared to endure that in order to keep following Christ? Many Christians have had some, if not all of their family reject them after they became Christ's followers. That has not stopped them following. Indeed, some folk in the Free Church use the phrase of a new Christian, "Oh, he (or she) has started following."
The simple application of what Jesus said is that, no matter what shame, disgrace or hatred comes to a follower of Jesus, they will grasp that as 'par for the course', to put it in modern terms. They will not shrink from any of that but identify with it as the cost of being a follower of Jesus.
First century Christians were later told in epistles to their new congregations that "the preaching of the cross [Christ's cross, that is] is to them that perish foolishness; but unto us which are saved it is the power of God" 1 Corinthians 1:18. Christians embrace the power of God - they embrace the cross - they identify with it, and with whatever personal affliction comes to them because they bear the name of Christ. They know that God blotted out the condemnation to sinners, taking "it out of the way, nailing it to the cross; having spoiled principalities and powers, he made a shew of them openly, triumphing over them in it" Colossians 2:14-15. Post-resurrection of Christ, believers saw the cross of Christ in that glorious light. No longer was it a defeat, a shame, a horror - in light of Christ's resurrection, it was the power of God unto salvation!
This highlights the second (and probably the most important) sense in which Christians pick up their cross. They proclaim the good news of what the cross of Christ achieved for sinners - their liberation from slavery to sin when they repent and embrace Christ's finished work on the cross. This is gospel, the good news, the evangel. And that is one sense in which evangelical Christians may take the meaning of Matthew 10:38 more boldly than do those who shrink from being called "Evangelical" Christians. To publicly proclaim the cross of Christ as the power of God unto salvation is to identify with the cross personally. That means testifying to how the cross of Christ smashed the chains of one's own slavery to sin. Maybe Evangelical Christians are more prone to do that than are Christians who don't care to be identified as 'Evangelical'. I don't know. I simply express this answer as a Christian who has been proud to identify with the cross of Christ for over four decades now, not being embarrassed at being labelled 'Evangelical'. That's fine by me!