question from a newbie: personalised map for private use

Yes. There are two general ways to do this.

One is to download the data, modify it, and use a rendering software to make a map from that modified data. This means you will have to install a suitable rendering software yourself. Enter "render my own map" in the search box above to look at some responses concerning this. You might want to try "Maperitive" as an easy entry into map rendering; it comes with a ready-made map style that looks similar to what you know from www.openstreetmap.org. This option has the advantage that you'll be able to even create tiles, i.e. a zoomable, pannable map.

The other option is to download a ready-made map and try to change the names on that. This is probably only feasible if you want to change just a few names, and you won't get a slippy map that way, but you don't have to install a renderer either. You can go to the "export" tab on www.openstreetmap.org and export a map in SVG format, then use an SVG editor like "Inkscape" to remove the letters you don't want and place others instead. This is going to be cumbersome because road labeling consists of lots of individual letters, rather than one text element.