Can Auto Layout constraints push content outside of the view in iOS Swift?

You’re not that far off… but couple of things:

The .tableHeaderView needs .translatesAutoresizingMaskIntoConstraints to be True, not False

Because you are using a multi-line label, we have to tell auto-layout how wide the view can be when requesting its "fitting" size – so viewDidLayoutSubviews() becomes:

override func viewDidLayoutSubviews() {
    super.viewDidLayoutSubviews()
    
    if let tableHeader = secondTableView.tableHeaderView {
        // define maximum width for the header view
        let fitSize: CGSize = CGSize(width: secondTableView.frame.width, height: .greatestFiniteMagnitude)
        // ask UIKit to give us the "fitting" size
        let sz: CGSize = tableHeader.systemLayoutSizeFitting(fitSize, withHorizontalFittingPriority: .required, verticalFittingPriority: .defaultLow)
        let height: CGFloat = sz.height
        var headerFrame = tableHeader.frame
        if height != headerFrame.size.height {
            headerFrame.size.height = height
            tableHeader.frame = headerFrame
            secondTableView.tableHeaderView = tableHeader
        }
    }
}

Here’s your controller class, with those slight modifications:

class SecondVC: UIViewController, UITableViewDelegate, UITableViewDataSource {
    
    @IBOutlet weak var secondTableView: UITableView!
    
    var stuff = [("Some other stuff 1"),("Some other stuff 2"),("Some other stuff 3")]
    var insertionText: String = "This is some sample insertion text that is currently over writing the first cell. Let's fix that."
    
    override func viewDidLoad() {
        super.viewDidLoad()
        secondTableView.delegate = self
        secondTableView.dataSource = self
        
        let tableHeader = UIView()
        secondTableView.tableHeaderView = tableHeader
        tableHeader.backgroundColor = .systemOrange

        // SHOULD NOT BE SET TO FALSE           
        //tableHeader.translatesAutoresizingMaskIntoConstraints = false
        //      let size = tableHeader.systemLayoutSizeFitting(UIView.layoutFittingCompressedSize)
        //      let height = size.height
        //      let width = size.width
        //      tableHeader.frame = CGRectMake(0, 0, width, height)
        
        let labelOne = UILabel(frame: tableHeader.bounds)
        labelOne.text = "USER:"
        labelOne.numberOfLines = 0
        labelOne.lineBreakMode = .byWordWrapping
        labelOne.textAlignment = .center
        labelOne.translatesAutoresizingMaskIntoConstraints = false
        
        let labelTwo = UILabel(frame: tableHeader.bounds)
        labelTwo.translatesAutoresizingMaskIntoConstraints = false
        labelTwo.text = "Inherited text here: (insertionText)"
        labelTwo.numberOfLines = 0
        labelTwo.lineBreakMode = .byWordWrapping
        labelTwo.textAlignment = .center
        labelTwo.translatesAutoresizingMaskIntoConstraints = false
        
        tableHeader.addSubview(labelOne)
        tableHeader.addSubview(labelTwo)
        
        // let's make the constraint setup a little more readable
        let g = tableHeader.layoutMarginsGuide
        
        NSLayoutConstraint.activate([
            
            labelOne.topAnchor.constraint(equalTo: g.topAnchor, constant: 11.0),
            
            labelOne.leadingAnchor.constraint(equalTo: g.leadingAnchor, constant: 20.0),
            labelOne.trailingAnchor.constraint(equalTo: g.trailingAnchor, constant: -20.0),
            
            labelTwo.topAnchor.constraint(equalTo: labelOne.bottomAnchor, constant: 0.0),
            
            labelTwo.leadingAnchor.constraint(equalTo: g.leadingAnchor, constant: 20.0),
            labelTwo.trailingAnchor.constraint(equalTo: g.trailingAnchor, constant: -20.0),

            labelTwo.bottomAnchor.constraint(equalTo: g.bottomAnchor, constant: -11.0),
            
        ])

        // so we can see the label frames
        labelOne.backgroundColor = .cyan
        labelTwo.backgroundColor = .yellow
    }
    
    override func viewDidLayoutSubviews() {
        super.viewDidLayoutSubviews()
        
        if let tableHeader = secondTableView.tableHeaderView {
            // define maximum width for the header view
            let fitSize: CGSize = CGSize(width: secondTableView.frame.width, height: .greatestFiniteMagnitude)
            // ask UIKit to give us the "fitting" size
            let sz: CGSize = tableHeader.systemLayoutSizeFitting(fitSize, withHorizontalFittingPriority: .required, verticalFittingPriority: .defaultLow)
            let height: CGFloat = sz.height
            var headerFrame = tableHeader.frame
            if height != headerFrame.size.height {
                headerFrame.size.height = height
                tableHeader.frame = headerFrame
                secondTableView.tableHeaderView = tableHeader
            }
        }
    }
    
    func tableView(_ tableView: UITableView, numberOfRowsInSection section: Int) -> Int {
        stuff.count }
    
    func tableView(_ tableView: UITableView, cellForRowAt indexPath: IndexPath) -> UITableViewCell {
        let cell = secondTableView.dequeueReusableCell(withIdentifier: "SecondTVCell", for: indexPath) as! SecondTVCell
        cell.someOtherStuffLabel.text = stuff[indexPath.row]
        return cell }
    
}

I find it very helpful during development to give UI elements contrasting background colors… makes it easy to see the framing at run-time. So I set the header view labels to cyan and yellow, and this is the result:

and when the device is rotated:

Note: when you run this, you will see some auto-layout complaints. That’s because we’re setting constraints on the labels and adding them (and the header view) to the view hierarchy in viewDidLoad() … but the constraints have conflicts at that point.

We can safely ignore those, but if we want to avoid them we can give the bottom constraint from the second label a priority of less-than required:

let g = tableHeader.layoutMarginsGuide
        
NSLayoutConstraint.activate([
            
    labelOne.topAnchor.constraint(equalTo: g.topAnchor, constant: 11.0),
            
    labelOne.leadingAnchor.constraint(equalTo: g.leadingAnchor, constant: 20.0),
    labelOne.trailingAnchor.constraint(equalTo: g.trailingAnchor, constant: -20.0),
            
    labelTwo.topAnchor.constraint(equalTo: labelOne.bottomAnchor, constant: 0.0),
        
    labelTwo.leadingAnchor.constraint(equalTo: g.leadingAnchor, constant: 20.0),
    labelTwo.trailingAnchor.constraint(equalTo: g.trailingAnchor, constant: -20.0),

    // don't set this here    
    //labelTwo.bottomAnchor.constraint(equalTo: g.bottomAnchor, constant: -11.0),
            
])

// this will satisfy auto-layout
let c = labelTwo.bottomAnchor.constraint(equalTo: g.bottomAnchor, constant: -11.0)
c.priority = .required - 1
c.isActive = true