Am not fond of heat-stripping enamel wire. Yes, a hot soldering iron can eventually strip off the enamel. Applying solder liberally helps this process. But a mess of burnt residue usually is left, which is best scrapped off after cooling. Then the wire end is best re-tinned a-fresh to shed excess solder coating. It is all too easy to apply insufficient heat to actually strip the enamel, but coat the wire with solder. Looks metallic, but not electrically connected to copper.
With heat-stripping enamel, I scrape through first. Maybe not all the way around, but enough to allow soldering iron to meet fresh copper. It strips much faster, as Alan G8LCO suggests. Burnt residue still requires scraping off. With older high-temperature enamel, a thorough,complete scraping first is absolutely required, and it is visibly apparent when you've missed a bit after tinning. No burnt residue is left. This kind of enamel requires more prep work, which means that nicking copper is a bigger risk. Both types of enamel requires scraping IMO. And tinning with fluxed solder is absolutely required IMO - the risk of remnant nearly-invisible enamel is too great: tinning exposes it. I'd rather put in a full scraping effort, and be assured that enamel is gone for sure, with a clear transition from tinned copper to insulating enamel - that favours the high-heat non-striping enamel.
One other thing - reusing old enamel wire is risky. Some windings are fixed in place with a bonding agent - when you remove enamel wire from such a form (like TV deflection scan coils) the enamel can de-bond from the copper. Really old enamel often becomes brittle - you can hear and see it crack and de-bond from the copper when flexed. Not appropriate for a tight toroid winding on small cores. In any case, if you re-wind a toroid, I'd recommend using fresh wire - the original winding has had its enamel stressed too far - original winding always goes to recycling.