I'm not an expert, the one that people recommend is exceedingly expensive at about £100 (and it doesn't have a mapping screen either but it is very good at what it is designed to do which is track points, lots and lots of them).
Your phone is probably fine for tagging or getting the idea. As long as it outputs GPX or something that can be converted into GPX it's fine.
There are plenty of other ways you can help out that don't involve doing tagging or tracks. You can source things through "local knowledge" or "surveys" on there. For example I added libraries, some shops, permissive paths and things and shedloads of cycle parking. Some places don't even have the road names on. There's billions of little bits and bobs people can do to slowly improve an area.
P.S. OSM updates almost instantaneously, whereas Opencyclemap is supposed to update once a week it is a bit more relaxed than that and can be up to every 2 weeks sometimes

.