I think that you are looking at this incorrectly. It doesn't matter what terminology you use. The existing sponsors know the terminology (or can employ people who do).
Just write your own Representative (plus the Speaker of the House and your two Senators) asking for support. Don't explain what kind of support you want. Just ask for support for that bill. If you convince them, they'll figure out what to actually do. If you don't convince them, it won't matter how accurate your terminology. They won't help you.
Representatives actively filter out campaigns from non-constituents. If you write Representatives outside your district, the staff will compare addresses and toss your letter. At best they might refer it to your Representative. You can see this if you try to email a Representative. They'll ask for your address and won't accept the email without it. And they won't accept the email with it if it is outside their district.
In some ways, accurate terminology may hurt your cause. Who knows accurate terminology? Other members of Congress. Who doesn't? Voters. So accurate terminology suggests an organized attempt by other members of Congress rather than a spontaneous feeling by a voter. And politicians care far more about what their voters think than their colleagues. Accurate terminology makes the letter less convincing and thereby less effective.
The other thing that you can do is find out who they think are their supporters and convince those people. Parties have local committees. Convince them and have them write the politicians. Convince their major donors and have them write the politicians. Convince the politicians that they will lose a primary, because their base won't support them without that bill.